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ReviewMay 5, 2025

Benchmarking Quantum Gates and Circuits
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Vinay Tripathi
Vinay Tripathi
Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, United States
Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States
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https://orcid.org/0000-0003-1418-9046
Daria Kowsari
Daria Kowsari
Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, United States
Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States
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Kumar Saurav
Kumar Saurav
Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States
Ming Hsieh Department of Electrical & Computer Engineering, University of Southern California, Los Angeles, California 90089, United States
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Haimeng Zhang
Haimeng Zhang
Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States
Ming Hsieh Department of Electrical & Computer Engineering, University of Southern California, Los Angeles, California 90089, United States
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Eli M. Levenson-Falk*
Eli M. Levenson-Falk
Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, United States
Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States
Ming Hsieh Department of Electrical & Computer Engineering, University of Southern California, Los Angeles, California 90089, United States
*E-mail: [email protected]
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https://orcid.org/0000-0002-0019-9465
Daniel A. Lidar*
Daniel A. Lidar
Department of Chemistry, University of Southern California, Los Angeles, California 90089, United States
Ming Hsieh Department of Electrical & Computer Engineering, University of Southern California, Los Angeles, California 90089, United States
Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, United States
Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States
*E-mail: [email protected]
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Chemical Reviews

Cite this: Chem. Rev. 2025, XXXX, XXX, XXX-XXX

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https://doi.org/10.1021/acs.chemrev.4c00870

Published May 5, 2025

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Abstract

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Accurate noise characterization in quantum gates and circuits is vital for the development of reliable quantum simulations for chemically relevant systems and fault-tolerant quantum computing. This paper reviews a variety of key benchmarking techniques, including Randomized Benchmarking, Quantum Process Tomography, Gate Set Tomography, Process Fidelity Estimation, Direct Fidelity Estimation, and Cross-Entropy Benchmarking. We evaluate each method’s complexities, the resources they require, and their effectiveness in addressing coherent, incoherent, and state preparation and measurement (SPAM) errors. Furthermore, we introduce Deterministic Benchmarking (DB), a novel protocol that minimizes the number of experimental runs, exhibits resilience to SPAM errors, and effectively characterizes both coherent and incoherent errors. The implementation of DB is experimentally validated using a superconducting transmon qubit, and the results are substantiated with a simple analytical model and master equation simulations. With the addition of DB to the toolkit of available benchmarking methods, this article serves as a practical guide for choosing and applying benchmarking protocols to advance quantum computing technologies.

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Subjects

Special Issue

Published as part of Chemical Reviews special issue “Quantum Computing”.

1. Introduction

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The realization of the promise of quantum computation (1,2) relies on the development of fault-tolerant, error-corrected quantum computers. Fault-tolerance is essential for the ultimate scalability of quantum computing, but requires high-fidelity operations that exceed an accuracy threshold. (3−7) When targeting chemical accuracy (1 kcal/mol) for electronic structure simulations, the impact of gate error becomes magnified because small inaccuracies in energies ─ especially for strongly correlated molecules ─ can alter reaction pathways and observables. Error probabilities per gate below 10^–4 could be vital in large-scale fault-tolerant simulations of industrially relevant catalysts, as higher error rates dramatically inflate the number of logical qubits and error-correcting cycles required to achieve a final result with the desired precision. (8) Similarly, in quantum dynamics calculations, which often involve iterated time-step evolution, the accumulated error can become prohibitively large unless physical error rates are suppressed to the 10^–5 – 10^–6 regime. (9) In these scenarios, each additional order of magnitude in gate fidelity can translate into disproportionately lower qubit counts or reduced circuit depth, thereby expanding the range of molecular systems amenable to quantum simulation.

Characterizing quantum operations is thus crucial and plays a pivotal role in the development and implementation of universal quantum computers. Accurate characterization allows for the identification and subsequent improvement of errors in quantum gates. But even before the advent of universal, fault tolerant quantum computation, today’s noisy intermediate scale quantum (NISQ) computers (10) are being actively investigated in terms of quantum chemistry applications. (11,12) Early algorithms focused on exact diagonalization, (13,14) while much current attention is devoted to variational algorithms. (15,16) From a control-theoretic perspective, reducing gate errors ─ particularly multiqubit errors that can amplify coherent leakage or correlated noise channels ─ is indispensable for accurate electronic structure calculations. (17) Near-term algorithms such as the Variational Quantum Eigensolver (VQE) (18) and, eventually, more resource-intensive protocols like Quantum Phase Estimation (QPE), (19) hinge on efficient gate decompositions that preserve delicate coherence over increasingly large circuit depths. (20) Any deviation in gate fidelity can accumulate into significant numerical biases, thus compromising predictions of reaction energies or catalytic mechanisms. (21) Consequently, noise-reduction strategies have emerged as key tools in the quest to push current hardware toward the threshold of chemical accuracy. (22) For such techniques to function and improve performance, accurate characterization and benchmarking of quantum operations are essential.

Several methods have been developed over the years for benchmarking quantum operations, particularly focusing on quantum gates. These methods are part of the broader research field of Quantum Characterization, Verification and Validation (QCVV), (23) and include Quantum Process Tomography (QPT), (24−26) Randomized Benchmarking (RB) and its many variants, (27−30) Gate Set Tomography (GST), (31−33) Process Fidelity Estimation, (34) Direct Fidelity Estimation, (35,36) and Cross-Entropy Benchmarking. (37−39) QPT provides a full reconstruction of a quantum process, but is resource-intensive and scales poorly with system size. RB, on the other hand, offers a scalable alternative by estimating the average error rate of quantum gates, but it lacks the ability to identify specific error types. GST addresses some of these limitations by providing a self-consistent characterization of a complete set of quantum gates, albeit at the cost of increased experimental complexity.

Gate errors are fundamentally either coherent or incoherent. Coherent errors that arise from systematic deviations in quantum gate operations pose a significant challenge in the context of error correction. (40,41) While stochastic errors add up as probabilities, coherent errors add up as amplitudes, generally leading to a quadratically smaller accuracy threshold. (42) Traditional error characterization methods such as RB, which deliberately randomize noise maps, do not capture coherent error accumulation unless appropriately modified. (43) This can result in an underestimation of the gate performance needed to achieve accuracy thresholds, critical for achieving fault-tolerant quantum computing. (44,45) Various techniques exist that can mitigate the accumulation of coherent errors by converting them into stochastic noise, but only at the expense of significant additional cost in the number of required circuits. (46−49) It can be advantageous to, instead, characterize coherent gate errors, (50) and subsequently eliminate or at least suppress such errors at the control level, rather than placing the burden of doing so on fault-tolerant quantum error correction using large codes. (51)

In this Review, we provide a survey of quantum gate characterization methods, emphasizing the role of each strategy depending on the end goal. As the field is vast, we sacrifice generality somewhat by emphasizing the characterization of single-qubit and two-qubit gates, given that these are the fundamental building blocks of quantum circuits. (52) However, we also include methods designed to provide holistic benchmarks at the algorithm level.

After providing, in Section 2, a general introduction to the sources of gate errors and the formalism for characterizing them, we cover in Section 3, a variety of methods ranging from the most comprehensive and resource-intensive, such as quantum process tomography and gate set tomography, to the least detailed, such as randomized benchmarking. We discuss the trade-offs between these techniques, their scalability, and their applicability to various quantum computing architectures.

To keep this review focused, we refrain from providing a comprehensive analysis of all available benchmarking methods. Consequently, we do not discuss several significant methods, such as cycle benchmarking, (53) leakage and crosstalk characterization, (54) quantum volume benchmarking, (55) fidelity estimation with 2-designs, (56) mirror benchmarking, (57) variational quantum benchmarking, (58) average gate fidelity, (59) machine learning-based benchmarking, (60,61) channel spectrum benchmarking, (62) and error per layered gate (EPLG), (63) among others. Most of these methods are variants of the key methods we focus on here, so we hope that the reader will feel well-prepared to tackle these other methods after reading this review.

However, this work is more than a review, as we also introduce an original benchmarking protocol that we call deterministic benchmarking (DB). This new protocol, which is appropriate for characterizing single-qubit gates, requires fewer experimental runs than previous characterization methods, exhibits robustness to SPAM errors, and provides detailed information about coherent errors. We discuss DB in detail in Section 4, provide experimental evidence of its efficacy, and contrast it with RB.

2. Gate Errors

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2.1. Quantum Dynamical Maps

Assuming a factorized initial system-bath state, i.e., a system which is not entangled with its bath, the dynamics of an open quantum system with a D-dimensional Hilbert space

H

can be described by a completely positive trace-preserving (CPTP) linear map: (64)

E (ρ) = \sum_{i} A_{i} ρ A_{i}^{†}

(1)

where

ρ \in B (H)

(the space of linear operators acting on

H

) is the initial state of the system and satisfies Tr(ρ) = 1 as well as ρ ≥ 0 (its eigenvalues are all non-negative). The Kraus operators (65) A_i satisfy ∑_iA_i^†A_i = I (the identity operator), which guarantees that

Tr [E (ρ)] = 1

. Let {E_m}_m=0^D²–1 be a set of fixed Hermitian basis operators for

B (H)

, which satisfy the orthogonality condition

Tr (E_{m}^{†} E_{n}) = D δ_{m n}

(2)

For example, we may choose E₀ = I and E_m, m = 1, ..., D² – 1, to be traceless Hermitian matrices. For a multiqubit system, the E_m’s can be tensor products of the identity operator and the Pauli matrices, which we denote interchangeably by {σ^x, σ^y, σ^z} or {X, Y, Z}. We also write Pauli tensor products interchangeably as σ^x ⊗ σ^y or XY, etc.

The A_i operators can be decomposed as A_i = ∑_ma_imE_m, and therefore we have

E (ρ) = \sum_{m, n = 0}^{D^{2} - 1} χ_{m n} E_{m} ρ E_{n}^{†}

(3)

where χ_mn = ∑_ia_in^*a_im = χ_nm^*. The “process matrix” χ is not only Hermitian but also positive semidefinite (all its eigenvalues are non-negative). It captures all the information about the map

E

with respect to the {E_i} basis, i.e., a characterization of

E

is equivalent to a determination of all the matrix elements of χ, where the E_i play the role of observables. Since χ is Hermitian and the map

E

is trace-preserving, the corresponding process matrix χ has at most D⁴ – D² independent real parameters [see eq 41 below]. We will focus most of our attention on the n-qubit case, i.e., D = 2ⁿ.

The measurement formalism of quantum mechanics is conveniently formulated using the language of CPTP maps as well. (66) A set of generalized measurement operators {A_i} is any set in

B (H)

that satisfies the constraint ∑_iA_i^†A_i = I. When a system in the state ρ is measured, the postmeasurement state is ρ_i′ = A_iρA_i^†/p_i with probability p_i = Tr(A_iρA_i^†) = Tr(M_iρ), where the set {M_i} = {A_i^†A_i} is known as a positive operator-valued measure (POVM). If the measurement outcome is not known, the resulting state becomes ∑_iA_iρA_i^†.

The standard case of projective von-Neumann measurements is the special case when the generalized measurement operators are projectors, i.e., are Hermitian and satisfy A_iA_j = δ_ijA_i. When an observable (any Hermitian operator) Ω with spectral decomposition Ω = ∑_kλ_kP_k (λ_k and P_k are its eigenvalues and eigenprojectors, respectively) is measured, the postmeasurement state is ρ_k′ = P_kρP_k/p_k with probability p_k = Tr(P_kρ), and the measurement outcome is the eigenvalue λ_k. The state resulting from ignorance of the measurement result is then ∑_kP_kρP_k, which generally corresponds to a mixed state, i.e., a state σ whose purity Tr(σ²) is less than 1.

The Choi-Jamiołkowski isomorphism (67,68) is a correspondence between maps acting on states in

B (H)

and states in

H \otimes H

. More concretely, the Choi state

ρ_{E}

of a quantum map

E

ρ_{E} = (I \otimes E) (| ϕ ⟩ ⟨ ϕ |)

(4)

where

| ϕ ⟩ = \frac{1}{\sqrt{D}} \sum_{i = 1}^{D} | i ⟩ \otimes | i ⟩

is a maximally entangled state. The Uhlmann fidelity (69) between two states

ρ, σ \in B (H)

F (σ, ρ) = ∥ \sqrt{ρ} \sqrt{σ} ∥_{1} = Tr ({(σ^{1 / 2} ρ σ^{1 / 2})}^{1 / 2})

(5)

where the norm is the trace-norm:

∥ A ∥_{1} = Tr | A | = Tr (\sqrt{A^{†} A}) .

The Uhlmann fidelity is bounded between 0 and 1 and is unitarily invariant. Note that some authors define it as the square of our quantity.

The process fidelity F between two quantum maps

E

and

E^{'}

, acting on the same quantum system, with respective Choi states

ρ_{E}

and

ρ_{E'}

, is given by the Uhlmann fidelity between the Choi states:

F_{proc} (E, E^{'}) = F (ρ_{E}, ρ_{E'}) = ∥ \sqrt{ρ_{E}} \sqrt{ρ_{E'}} ∥_{1}

(6)

F_proc quantifies the overlap between the two maps’ actions on all possible input states. A process fidelity value close to 1 indicates that the maps are very similar in their effect on quantum states. This metric is very useful for quantifying the similarity between a desired process, such as an ideal gate, and its actual, noisy implementation.

2.2. Coherent and Incoherent Errors

A quantum gate aims to produce some unitary operation U which can be written as a superoperator

U

acting on the state ρ, i.e.,

U (ρ) = U ρ U^{†}

. In practice, the actual operation is

\tilde{U} = E ◦ U

, where

E

now represents all errors on the gate. These errors can be divided into coherent errors, where state purity is preserved but the unitary is altered, and incoherent (stochastic) errors, where state purity [Tr(ρ²) = 1] is lost due to some randomness in the evolution (i.e., decoherence). Put another way, coherent errors are deterministically repeatable, while incoherent errors give random evolution. Coherent errors are generally more damaging, as their amplitudes add in phase and can thus increase the gate failure probability quadratically faster than incoherent errors. However, coherent errors are always reversible in principle. Incoherent errors, in contrast, require quantum error correction to become reversible. (70) Next, we present an overview of common incoherent and coherent errors in single- and two-qubit gates.

2.3. Single-Qubit Gate Errors

In general the map

E

is very unconstrained. For a single qubit, i.e., for D = 2, this allows D⁴ – D² = 12 real parameters in a CPTP map, and even more if we allow for leakage outside the qubit space (see below). On the other hand, the physical causes and impacts of error maps are more constrained. For a single qubit, the state ρ can be represented by the Bloch vector v ≡ (⟨σ_x⟩, ⟨σ_y⟩, ⟨σ_z⟩), with the length of the vector giving the state purity. A general single-qubit unitary can be parametrized as R(ϑ, θ, ϕ), a rotation by an angle ϑ about an axis with polar and azimuthal angles θ, ϕ. Coherent errors in the space of qubit states can then be categorized as

• Rotation errors, where the value of ϑ is changed from the target. These may be the result of drifts in the amplitudes of control drives or the coupling of the qubit to these drives.

• Phase errors, where the axis of rotation is altered─that is, θ and/or ϕ are changed from the target. These may result from drifts in the qubit frequency leading to unwanted z-axis rotations, or drifts in the control axis (often caused by a drift in the phase of an oscillatory control field).

Similarly we can characterize incoherent errors as due to

Relaxation from the qubit state to the equilibrium state. These bit-flip errors are typically modeled as Markovian processes leading to an exponential decay with characteristic time T₁. Importantly, relaxation at finite temperature leads to a nonzero state purity as the system relaxes toward its ground state.
Dephasing due to noise in the qubit frequency relative to a reference (where the noise may originate from the qubit or the reference). These phase-flip errors may have both Markovian and non-Markovian components, and, in many condensed matter or solid-state systems, are typically modeled as a Markovian error plus a classical noise with a 1/f power spectrum. (71)

It is also important to consider leakage errors, which result from the state evolving outside the qubit space. This error is coherent as long as it preserves state purity (thus allowing for its reversal (72)), although it will always appear incoherent if we restrict ourselves to only considering the qubit space. This is the case in, e.g., the transmon superconducting qubit, (73) where the qubit is encoded in the lowest two states of a weakly anharmonic oscillator. Driving this transition can also drive higher transitions, leading to coherent rotations in these state manifolds. Importantly, repeated application of a gate with a coherent leakage error can cause a rapid buildup of errors that can be difficult to address.

Modeling leakage in the qubit space requires a non-trace-preserving map, as population leaves the space and is transferred to the leakage state. Such maps can be modeled by modifying the CPTP map formalism (Section 2.1) and replacing the equality in the ∑_iA_i^†A_i = I constraint by ≤ I. The corresponding process matrix χ then has at most D⁴ independent real elements (16 for a single qubit).

A final error type, erasure, is a special type of leakage error where qubit information is lost with some characteristic signature that an error has occurred. This is the case in, e.g., dual-rail qubit encodings where single-excitation loss takes both qubit states to a single leakage state and thus erases all information, (74,75) or trapped atom qubits where an atom may escape the trap. (76) Because erasure scrambles qubit information it must be considered as an incoherent error. Some error correction architectures can correct erasure errors with far fewer resources than bit-flip or phase-flip errors. It can therefore be preferable to use encodings which transform other errors into erasures, (77) as is done with dual-rail qubits. On the other hand, in many common qubits such as the transmon, erasure errors are not present and leakage must be considered as a coherent effect. (78,79)

Incoherent errors are sometimes lumped together and modeled as a single depolarizing map

E (ρ) = (1 - p) ρ + \frac{1}{3} p (X ρ X + Y ρ Y + Z ρ Z)

(7)

where p is the probability of a qubit error. However, this both obscures their physical origin and does not account for the effects of biased noise (80−86) and finite temperature (see Section 4.8).

If we ignore detection of erasure errors, and limit ourselves to Markovian decoherence, we see 7 parameters for qubit gate error: errors in ϑ, θ, ϕ, relaxation rate, equilibrium population, dephasing rate, and leakage rotation. Some of these parameters (e.g., equilibrium population) may be independent of the gate itself, and so need not be characterized for every gate. Proper treatment of non-Markovian errors can be much more complicated (87,88) and depends on the circumstance, but can often be captured with relatively simple models. (71,89,90)

2.4. Two-Qubit Gate Errors

The set of two-qubit gates is much larger and more diverse than the simple rotation gate for a single qubit, though it is amenable to an elegant geometrical classification. (91) It is therefore more practical to consider the error maps for a particular two-qubit operation, and even for a particular physical implementation of that operation, rather than for general two-qubit operations. For instance, consider the parametrically activated iSWAP gate in transmon qubits: (92) an oscillatory flux drive causes oscillations of a tunable transmon’s frequency about some maximum, bringing the average frequency into resonance with another transmon so the two can exchange energy (possibly via an intermediate tunable transmon coupler). (93) Error maps include Markovian relaxation and dephasing of each qubit; non-Markovian dephasing of each qubit, typically modeled as a 1/f qubit frequency noise spectrum (note that the oscillatory nature of the drive acts as an “auto-echo” that eliminates much of this dephasing); two-qubit couplings to a Markovian or non-Markovian bath, e.g., a noisy tunable coupler; incorrect drive amplitude leading to transmon-transmon detuning and so causing incomplete swapping; two-qubit phase errors from unwanted ZZ interaction due to higher transmon levels; single-qubit phase errors due to improper calibration of level repulsion during transients at the beginning and end of the gate; and leakage into higher transmon levels or coupler states. Every other implementation of iSWAP can have its own error maps, to say nothing of other two-qubit operations.

The complexity and diversity of two-qubit gate error maps mean it is best to approach them holistically on a case-by-case basis, e.g, by developing an “error budget” for the different sources of error. (93,94) Another common approach is to characterize the gate completely once, using quantum process tomography or some other complete method. This identifies relevant error maps, which may then be measured more efficiently without needing to fully characterize the quantum process over and over. Typically the relevant coherent errors are characterized via some form of matrix element amplification, where repeated applications of the gate on a particular initial state build up a particular coherent error. There is a danger that this amplification will also amplify single-qubit coherent errors that may be difficult to distinguish. Adding dynamical decoupling sequences in between the two-qubit gate applications can help suppress these single-qubit errors that may confound the analysis. (50)

Incoherent errors in two-qubit gates are typically characterized by performing single-qubit coherence measurements, then using some error model (usually Markovian) to predict the impact of single-qubit decoherence. Unfortunately, two-qubit incoherent errors due to joint bath couplings are often ignored unless full process tomography is being performed. This simplification can be shown to give good predictions in many contexts, (95) but in some circumstances two-qubit incoherent errors may be quite significant. For instance, the two qubits are often coupled via a third system (such as a tunable coupler) that can act as a joint bath. Fortunately, it is straightforward to extend standard single-qubit coherence measurements─for instance, joint relaxation can be characterized by comparing the relaxation rates of the |10⟩, |01⟩, and |11⟩ states. (96)

Properly benchmarking two-qubit gates is a significant challenge, and the best approach depends on the exact goal: calibrating a gate, predicting algorithmic performance, characterizing the overall quality of a quantum processor, etc. We describe and compare the various single- and two-qubit gate benchmarking protocols below.

3. Gate and Circuit Benchmarking Protocols

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A gate benchmarking protocol produces a set of numbers for an input gate or gate set, which can be used to compare the performance in different instances of gate implementation. Such a protocol can be judged on the basis of a few main criteria:

• Comprehensiveness, i.e., how much information about the error does the protocol provide. One simple way to quantify the amount of information provided is the number of parameters that end up describing the error.

• Specificity, i.e., what assumptions are made about the noise model in the protocol. Some common assumptions are that errors are independent of each other and weakly depend on the gate being applied, (97,98) or that they are of a particular type, such as Pauli errors. (99)

• Simplicity, i.e., how easily the scheme can be adapted to existing experimental setups. This includes how complicated the pulse control and measurement schemes are for the experiment as well as how sensitively the experiment must be calibrated.

• Resource costs, i.e., how expensive it is to run the protocol. Note that scalability, which is a critical consideration in quantum algorithms and processes, can become less relevant for a gate benchmarking protocol if we restrict the characterization effort to single one- or two-qubit gates in isolation. The number of such gates is at most n(n – 1)/2 for n fully connected qubits. Instead, the absolute resource cost of the protocol becomes important and needs to be optimized.

Apart from these criteria, a gate benchmarking protocol should also be robust against all errors, apart from the ones it is trying to quantify. Critically, for most protocols, this robustness should include state preparation and measurement (SPAM) errors, (100) which are among the most pervasive errors in experiments. (101) Exceptions to this rule include protocols that quantify the SPAM errors themselves.

3.1. Randomized Benchmarking

Randomized Benchmarking (RB) aims to give the most concise characterization of gate error by reporting a single number to quantify the average gate fidelities, while suppressing the effects of SPAM errors. (27−29,102) Over the past two decades, RB has become a standard tool used across different platforms to quantify the performance of single- and two-qubit gates. Many variants of RB have been introduced; here, we focus on the two most prominent variants, known as standard RB and interleaved RB. The main benefits of RB methods include reasonable assumptions, simplicity of implementation compared to process tomography, and relatively low resource cost. However, RB reduces the characterization to a single number. Indeed, this is the main reason for the method’s relative simplicity.

3.1.1. Standard RB

Standard RB characterizes a specific selection of gates, which are elements of the Clifford group (see below), by an average gate infidelity metric. The key idea behind standard RB is to transform errors into an effective depolarizing error [recall eq 7] via a process called “twirling”, which simplifies the analysis. The process is described in detail below.

The Pauli group

P_{n}

is the group of all possible n-fold tensor products of the three Pauli matrices and the identity matrix. The Clifford group consists of the set of gates that preserve the Pauli group under conjugation. I.e., C is in the Clifford group if

C P C^{†} \in P_{n}

for all

P \in P_{n}

The standard RB protocol involves the following steps:

Preparation: Initialize the n-qubit quantum system in a reference state |0⟩^⊗n, typically the ground state.
Random Gate Sequence: Apply a sequence of m randomly selected Clifford gates {C_i}_i = 1^m:
$U_{m} = C_{m} C_{m - 1} \cdot\cdot\cdot C_{1}$
(8)
Inversion Gate: Apply the inverse of the accumulated operation to ideally return the system to the initial state:
$C_{inv} = C_{m + 1} = U_{m}^{†} = C_{1}^{†} \cdot\cdot\cdot C_{m}^{†}$
(9)
The inverse can be calculated efficiently classically. (103)
Measurement: Measure the system in the computational basis (the eigenbasis of the Pauli σ_z matrix) and record whether it has returned to the initial state.
Averaging: Average the measurement outcomes over K different random sequences for a given m to obtain the survival probability P(m). Repeat the process for various sequence lengths m to get a sequence of survival probabilities {P(m)}.
Fitting to exponential: The final step is to fit the measured survival probability to an exponential function Ap^m + B in order to extract p, which can be used to quantify the average gate fidelity of the Clifford gate sequences. Here, A and B capture the SPAM errors.

3.1.2. RB Theory

A randomly selected Clifford gate C can be represented by the ideal superoperator

G (\cdot) = C (\cdot) C^{†}

, and the associated error process is modeled as

\tilde{G} = E ◦ G

, where the error

E

is assumed to be deterministic. The total process map

{\tilde{I}}_{m}

for the sequence of m + 1 Clifford gates (including the last one which inverts the previous m gates) is given by

{\tilde{I}}_{m} = {\tilde{G}}_{m + 1} ◦ {\tilde{G}}_{m} ◦ \cdot\cdot\cdot ◦ {\tilde{G}}_{1}

(10)

Ideally,

{\tilde{I}}_{m}

should have been the identity operation. Assuming gate-independent noise i.e.,

E_{i} \equiv E \forall i

, we obtain

{\tilde{I}}_{m} = (E ◦ G_{m + 1}) ◦ (E ◦ G_{m}) ◦ \cdot\cdot\cdot ◦ (E ◦ G_{1})

(11)

which can be written as

{\tilde{I}}_{m} = E ◦ ◯_{i = 1}^{m} (D_{i}^{- 1} ◦ E ◦ D_{i})

(12)

where

D_{i} = \prod_{k = 1}^{i} G_{k}

and we have used

Π_{k = 1}^{m + 1} G_{k} = I

. This is the process generated by one of the realizations; repeating the same depth-m sequence K times results in an average process defined as

\begin{matrix} {\bar{I}}_{m} \equiv \frac{1}{K} \sum_{j} {\tilde{I}}_{m}^{(j)} \\ = E ◦ \frac{1}{K} \sum_{j} ◯_{i = 1}^{m} (D_{i} {(j)}^{- 1} ◦ E ◦ D_{i} (j)) \end{matrix}

(13)

Here the error map

E

is “twirled” over the Clifford group, and it can be shown (104) that this reduces it to an m-fold depolarizing map

E_{dep}^{◦ m}

which replaces a state ρ by the maximally mixed state I/D with probability 1 – p^m, and leaves the state unchanged with probability p^m. We note that this process of applying random unitaries that leave only a Markovian contribution is closely related to the earlier idea of randomized dynamical decoupling. (105−107)

The average survival probability of the initial state ρ₀ = (|0⟩⟨0|)^⊗n is then given by

P_{avg} (m) = Tr [ρ_{0} {\bar{I}}_{m} (ρ_{0})] = Tr [ρ_{0} E ◦ E_{dep}^{◦ m} (ρ_{0})] = Tr [ρ_{0} E (p^{m} ρ_{0} + (1 - p^{m}) \frac{I}{D})] = A p^{m} + B

(14)

where

A = Tr [ρ_{0} E (ρ_{0} - \frac{I}{D})]

(15)

and

B = Tr [ρ_{0} E (\frac{I}{D})]

(16)

The key assumptions include gate-independent and time-independent errors in the above derivation. P_avg(m) from eq 14 is used to fit the experimental decay curves to extract p. Since twirling generates a depolarizing noise map of the form

E_{dep} (ρ) = p ρ + (1 - p) \frac{I}{D}

, its average gate fidelity with respect to an identity can be calculated to be

{\bar{F}}_{E_{dep}, I} = p + (1 - p) / D

, where the depolarization parameter p is identified with the experimentally measured exponential decay parameter. Therefore, we can relate the average Clifford gate infidelity

r_{C} = 1 - {\bar{F}}_{E_{dep}, I}

to the experimental decay parameter p obtained from the RB experiments as

r_{C} = \frac{D - 1}{D} (1 - p)

(17)

Since r_C depends only on the depolarization parameter p of the effective process matrix generated by the gate, RB becomes insensitive to SPAM errors, which are captured by A and B.

If we consider the case when the errors are gate-dependent as well as time-dependent, a perturbation theory calculation can be performed to extract the higher order corrections. In particular, the first order correction yields (29,102)

P_{avg}^{(1)} (m) = A_{1} p^{m} + B_{1} + C_{1} (m - 1) (q - p^{2}) p^{m - 2}

(18)

where A₁, B₁ and q are functions of m. The term (q – p²) is a measure of the degree of gate-dependence in the errors. It can be shown that the second order term can be neglected when γ ≪ 2/m, where γ is a measure of the noise variation. (29) The zeroth order approximation [see eq 14] provides a good estimate for small errors and the first order correction may be needed in cases where the errors are large. Note that even if the gate dependence term q – p² is large, it would still be possible to fit the experimental data with eq 18 as long as m is large enough. However, this can conflict with the γ ≪ 2/m condition. In addition, it is also necessary for m to be large enough to have enough data points for a reasonable estimate of p in the fitting model. These considerations highlight some of the restrictions of using RB for gate and time-dependent noise.

3.1.3. Interleaved RB

Whereas RB provides an averaged fidelity for a random set of Clifford gates, Interleaved RB (IRB) (108) allows the estimation of the fidelity of a specific Clifford gate

C

. The key idea is to analyze the effect of inserting the gate of interest G (represented below by a superoperator

G

) into an RB sequence. The IRB protocol consists of two different experiments: the first is the standard RB as discussed in Section 3.1.1, while the second is an RB sequence with

G

interleaved between every pair of consecutive random Clifford gate. More concretely, the sequence of gates in the second experiment after interleaving is

U_{m, inter} = G_{m} ◦ G ◦ G_{m - 1} ◦ \cdot\cdot\cdot ◦ G ◦ G_{2} ◦ G ◦ G_{1}

(19)

The above sequence is then followed by

G_{m + 1} = U_{m, inter}^{- 1}

. Similar to standard RB, this also generates an exponential decay characterized by a parameter p_inter. It can then be shown (108) that using the two decay parameters p and p_inter from the two experiments, we can estimate the gate infidelity contribution of the particular gate

G

using

r_{G} = \frac{D - 1}{D} (1 - \frac{p_{inter}}{p})

(20)

We note that IRB is not always a reliable method for estimating a gate’s infidelity because unitary errors in the interleaved gate

G

can coherently add or cancel with errors in the random Clifford gates, sometimes even resulting in a negative error rate. (23) Another way to see the problem with IRB is via eq 20 with the interleaved decay slower than the decay of standard RB (p_inter > p), which results in a negative infidelity

r_{G} .

This, in turn, implies an unphysical average gate fidelity larger than 1. Such issues lead to potentially large discrepancies between the estimated and true fidelity.

In fact, it can be shown (108) that the original error rate for the Clifford gate of interest

G

must lie in the range

[r_{G} - E, r_{G} + E]

where

E = \min {\begin{matrix} \frac{(D - 1) [| p - p_{inter} / p | + (1 - p)]}{D} \\ \frac{2 (D^{2} - 1) (1 - p)}{p D^{2}} + \frac{4 \sqrt{1 - p} \sqrt{D^{2} - 1}}{p} \end{matrix}

(21)

which arises from imperfect random gates. When the noise channel is fully depolarizing (p = 1) then E = 0, but in general E > 0. This means that

r_{G} + E

can exceed 1, which is unphysical. When this happens, IRB results must be treated with caution. Possible remedies include increasing the number of random sequences and the number of circuit repetitions, and checking for systematics in how errors are introduced (e.g., calibration or readout errors might be overshadowing gate error in unexpected ways).

Despite its shortcomings, randomized benchmarking is a fundamental tool in quantum computing for quantifying average gate fidelities with robustness to SPAM errors. By leveraging the mathematical properties of the Clifford group and statistical averaging, RB provides a scalable method for characterizing quantum processors. Ongoing research continues to refine RB protocols and extend their applicability to more general settings. IRB is just one possible extension of RB; there is a plethora of other work focusing on other generalizations of randomized benchmarking. These include simultaneous RB (109) (which detects the effects of simultaneous gate operations), leakage RB, (110,111) non-Markovian RB, (112,113) mirror RB, (114) character RB (115,116) (which goes beyond the Clifford group), and more.

3.2. Quantum Process Tomography

If randomized benchmarking sacrifices some comprehensiveness and specificity in service of simplicity and low resource cost, quantum process tomography (QPT) does the opposite. This method seeks to fully characterize a quantum process

E

, instead of providing a single number as in RB. Since D⁴ – D² real numbers are required to fully specify a CPTP map

E

, QPT is, of course, much more resource intensive than RB.

3.2.1. Standard Quantum Process Tomography

In “standard” quantum process tomography (SQPT), illustrated in Figure 1(a), we prepare D² linearly independent inputs {ρ_k}_k=0^D²–1, where each

ρ_{k} \in B (H)

, and then measure the output states

E (ρ_{k})

using quantum state tomography. (24,25) Since the map

E

[eq 1] is linear, it can be in principle reconstructed from the measured data by proper inversion. Let {ρ_k}_k=0^D²–1 be a linearly independent basis set of operators for

B (H)

. A convenient choice is ρ_k = |m⟩⟨n|, where {|m⟩}_m=0^D–1 is an orthonormal basis for

H

and k = (m, n). The coherence |m⟩⟨n| can be constructed from four populations:

\begin{matrix} | m ⟩ ⟨ n | = | +_{m n} ⟩ ⟨ +_{m n} | + | -_{m n} ⟩ ⟨ -_{m n} | \\ - \frac{1 + i}{2} (| m ⟩ ⟨ m | + | n ⟩ ⟨ n |) \end{matrix}

where

| \pm_{m n} ⟩ = (| m ⟩ \pm | n ⟩) / \sqrt{2}

. The linearity of

E

then implies that a measurement of

E (| +_{m n} ⟩ ⟨ +_{m n} |)

E (| -_{m n} ⟩ ⟨ -_{m n} |)

E (| m ⟩ ⟨ m |)

, and

E (| n ⟩ ⟨ n |)

suffices for the determination of

E (| m ⟩ ⟨ n |)

. In addition, the channel output

E (ρ_{k})

itself can be expressed in terms of a linear combination of basis states, as

E (ρ_{k}) = \sum_{l} λ_{k l} ρ_{l}

, where the parameters

λ_{k l} = Tr (ρ_{k} E (ρ_{l}))

(22)

contain the measurement results. Similarly, each summand in the RHS of eq 3 can be expanded in the basis states as E_mρ_kE_n^† = ∑_lB_mn,lkρ_l. The resulting equation then reads

\begin{matrix} E (ρ_{k}) = \sum_{m n l} χ_{m n} B_{m n, l k} ρ_{l} = \sum_{l} λ_{k l} ρ_{l} \\ \Leftrightarrow \sum_{m n} B_{m n, l k} χ_{m n} = λ_{k l} \end{matrix}

(23)

where the second line follows from the linear independence of {ρ_k}. This relation can in turn be written in the following matrix form:

B χ = λ

(24)

where the D⁴ × D⁴-dimensional matrix B is determined by the choice of operator bases {ρ_k} and {E_m}, and the D⁴-dimensional vector λ is determined from state tomography experiments. The process matrix χ can thus be determined by inversion of eq 24.

Figure 1. Three different QPT methods. (a) Schematic of SQPT. An ensemble of states {ρ_j} is prepared and each state is subjected to the map $E$ followed by measurements {E_m}. (b) Schematic of separable AAPT. A “faithful” input state ρ is subjected to the map $E \otimes I$ . The operators {E_j} are measured on the joint system, which results in the required joint probability distributions or expectation values. (c) Schematic of DCQD. The system and the ancilla are prepared in one of the input states as in Table 1 and are subjected to the map $E \otimes I$ . The joint system is measured in the Bell-state basis (Bell State Measurement).

In general, SQPT involves the preparation of D² linearly independent inputs {ρ_l}, each of which is subjected to the quantum process

E

, followed by quantum state tomography on the corresponding outputs. For each ρ_l, we must measure the expectation values of the D² fixed-basis operators {E_k} in the state

E (ρ_{l})

. Thus, the total number of measurements required is D⁴. Since measurement of an expectation value cannot be done on a single copy of a system, whenever we use the term “measurement” we implicitly mean measurement on an ensemble of identically prepared quantum systems corresponding to a given experimental setting.

3.2.2. Ancilla-Assisted Process Tomography

There is a duality between quantum state tomography schemes and QPT. This duality is based upon the Choi-Jamiolkowski isomorphism (67,68) we discussed in Section 2.1, which establishes a correspondence between CP maps and states. This one-to-one correspondence enables all theorems about quantum maps to be directly derived from those of quantum states. (117) In particular, it allows one to represent a quantum process as a quantum state in a larger Hilbert space. For example, choosing {E_m} = {|i⟩⟨j|} results in

χ = D ρ_{E}

, where

ρ_{E}

is the Choi state of the CPTP map

E

. Therefore, QPT has a natural reduction to quantum state tomography, and state identification methods can be applied to the characterization of quantum processes as well. The “ancilla-assisted process tomography” (AAPT) (118−120) scheme is based on this observation and is schematically illustrated in Figure 1(b).

The AAPT scheme involves attaching an ancilla B to the principal system A and preparing the combined system in a particular joint state so that complete information about

E

can be imprinted on the final state. Then, by performing quantum state tomography in the extended Hilbert space

H_{A B} = H_{A} \otimes H_{B}

, one can extract complete information about the unknown map

E

acting on A. In principle, the input state of the system and ancilla can be prepared in either an entangled mixed state (entanglement-assisted) or a separable mixed state. The only requirement for the input state in AAPT is that it must be “faithful” to the map

E

, i.e., performing quantum state tomography on the output identifies

E

completely and unambiguously. (121) The faithfulness condition implies that a state ρ can be used as input for AAPT iff it has maximal Schmidt number D², which is essentially an invertibility condition. By performing state tomography on the output state

(E \otimes I) (ρ)

, we obtain a set of linear relations among possible measurement outcomes and the elements of

E

and ρ; the choice of ρ must be such that an inversion is possible.

Maximal entanglement is not a necessary property of the input state ρ in AAPT. However, there is experimental evidence that maximally entangled pure states offer optimal performance in the sense of reducing experimental error. (120) Indeed, experimental error amplification is related to inversion of the matrix involving

E

and ρ, which, in turn, depends on the magnitude of the inverse of the eigenvalues of ρ. Thus, a smaller eigenvalue leads to an amplified experimental error. As a consequence, the optimal (in the sense of minimal experimental errors) faithful input states are maximally entangled pure states.

One can realize the required quantum state tomography via separable measurements (separable AAPT), i.e., joint measurement of tensor product operators, or via collective measurements on both the system and the ancilla (nonseparable AAPT). Both types of measurements are performed on the same Hilbert space

H_{A B}

. Furthermore, by adding another ancilla it is possible to perform a generalized measurement on the joint system AB. It turns out that the nonseparable measurement schemes (whether in the same Hilbert space or in a larger one) hardly offer any practical advantage because they require many-body interactions which are experimentally unrealizable. (26) We thus focus on joint separable measurements.

Let us assume that the initial state of the system and the ancilla is ρ_AB = ∑_ijρ_ijE_i^A ⊗ E_j^B, where {E_m^A} ({E_n^B}) is the operator basis for the linear operators acting on

H_{A}

(

H_{B}

). The output state, after the action of the unknown map

E

on the principal system, is

ρ_{A B}^{'} = (E_{A} \otimes I_{B}) (ρ_{A B}) = \sum_{k j} {\tilde{α}}_{k j} E_{k}^{A} \otimes E_{j}^{B}

(25)

where α̃_kj ≡∑_mniχ_mnρ_ijα_k^m,i,n, with α_k^m,i,n defined via E_m^AE_i^AE_n^A† = ∑_kα_k^m,i,nE_k^A. Note that α_k^m,i,n depends only on the choice of operator basis. The result of measuring the observables E_k^A ⊗ E_j^B is

{\tilde{α}}_{k j} = Tr (ρ_{A B}^{'} {(E_{k}^{A} \otimes E_{j}^{B})}^{†})

(26)

Now, let

{\tilde{χ}}_{k i} = \sum_{m n} α_{k}^{m, i, n} χ_{m n}

(27)

where the process matrix χ is the primary objective of QPT [recall eq 3] and the α parameters are known from the choice of operator basis. Thus, the procedure to obtain χ simplifies to two matrix inversions: after measuring {α̃_kj}, we obtain χ̃ by inverting the following matrix equation:

\tilde{α} = \tilde{χ} ϱ

(28)

Using χ̃, χ is then found by inverting eq 27.

3.2.3. Direct Characterization of Quantum Dynamics

An alternative to SQPT and AAPT is “direct characterization of quantum dynamics” (DCQD). (26,122−126) In DCQD, similarly to AAPT, an ancilla system is utilized, but in stark contrast, it does not require inversion of a D² × D² matrix; hence, “direct”. The main idea in DCQD is to use certain entangled states as inputs and to perform a simple error-detecting measurement on the joint system-ancilla Hilbert space. The choice of input states and measurements is such that the measurement results directly encode elements of the quantum map, which removes the need for state tomography. In the case of a single qubit, the measurement scheme turns out to be equivalent to a Bell state measurement (BSM). In DCQD the choice of input states is dictated by whether diagonal or off-diagonal elements (quantum dynamical population and coherence, respectively) of the process matrix are to be determined. Population characterization requires maximally entangled input states, while coherence characterization requires nonmaximally entangled input states. We review the scheme for the case of a single qubit below.

Let us first demonstrate how to determine all diagonal elements of the process matrix, {χ_mm}, in a single (ensemble) measurement. We choose {I, X^A, Y ^A, Z^A} as our operator basis acting on the principal qubit A. We first prepare the maximally entangled Bell state

| Φ^{+} ⟩ = {(| 00 ⟩ + | 11 ⟩)}_{A B} / \sqrt{2}

, and then (as in AAPT) subject only qubit A to the map

E

A stabilizer code is a subspace

H_{C}

of the Hilbert space of n qubits that is an eigenspace of a given Abelian subgroup

S

that excludes −I (the stabilizer group) of the n-qubit Pauli group

P_{n},

with eigenvalue +1. (127) In other words, for every

| Ψ_{C} ⟩ \in H_{C}

and all

S_{i} \in S \subset P_{n}

, we have S_i|Ψ_C⟩ = |Ψ_C⟩, where [S_i, S_j] = 0 ∀i,j. Consider the action of an arbitrary (error) operator

E \in P_{n}

on a stabilizer code state |Ψ_C⟩. Such an error is detectable if it anticommutes with (at least one of) the stabilizer generators: {S_i, E} = 0. To see this note that

S_{i} (E | Ψ_{C} ⟩) = - E (S_{i} | Ψ_{C} ⟩) = - (E | Ψ_{C} ⟩)

(29)

i.e., E|Ψ_C⟩ is a –1 eigenstate of S_i. Hence, measuring S_i detects the occurrence of an error or no error (the –1 or +1 outcomes respectively). Measuring all the stabilizer generators then yields a list of errors (the “syndrome”), which allows one to detect the errors that took place. More precisely, each stabilizer code has a distance parameter d. The code detects all errors whose weight (number of nonidentity operators) is at most d – 1. E.g., a distance d = 3 code can detect all one- and two-body errors, i.e., errors of the form XII, IYZ, etc., but not weight-3 errors such as XYZ.

The Bell state |Φ⁺⟩ is a +1 eigenstate of the commuting stabilizer generators Z^AZ^B and X^AX^B, i.e., it is stabilized under their action. The nonidentity error operators in the operator basis set {E_m^A}_m=0³ = {I, X^A, Y^A, Z^A} anticommute with at least one of the stabilizer generators, and therefore by measuring these generators we can detect the error. Note that, as mentioned above, measuring Z^AZ^B and X^AX^B is equivalent to a BSM: it can be represented by the four projection operators P_Ψ^± = |Ψ^±⟩⟨Ψ^±| and P_Φ^± = |Φ^±⟩⟨Φ^±|, where

| Φ^{\pm} ⟩ = (| 00 ⟩ \pm | 11 ⟩) / \sqrt{2}

, and

| Ψ^{\pm} ⟩ = (| 01 ⟩ \pm | 10 ⟩) / \sqrt{2}

are the Bell states. Expressing the map on qubit A as

E (ρ) = \sum_{m, n} χ_{m n} E_{m}^{A} ρ E_{n}^{A †}

, it is easily verified that the probabilities of obtaining the no-error outcome I and the errors X^A, Z^A, and Y^A can be written as

p_{m} = Tr [P_{m} (E \otimes I) (| Φ^{+} ⟩ ⟨ Φ^{+} |)] = χ_{m m}

(30)

where m = 0, 1, 2, 3, and the projectors P_m, for m = 0, 1, 2, 3, correspond to the states Φ⁺, Ψ⁺, Ψ^–, and Φ^–, respectively. For example:

\begin{matrix} Tr [P_{0} (E \otimes I) (| Φ^{+} ⟩ ⟨ Φ^{+} |)] \\ = \sum_{m, n = 0}^{3} χ_{m n} Tr [P_{Φ^{+}} (E_{m}^{A} \otimes I) P_{Φ^{+}} (E_{n}^{A †} \otimes I)] \\ = \sum_{m, n = 0}^{3} χ_{m n} δ_{m 0} δ_{n 0} = χ_{00} \end{matrix}

(31)

Eq 30 shows that the diagonal elements of the process matrix are directly obtainable from an ensemble BSM; this is the core observation that leads to the DCQD scheme. In particular, one can determine the quantum dynamical populations, χ_mm, by simultaneously measuring the operators Z^AZ^B and X^AX^B on multiple copies of the state |Φ⁺⟩.

To determine the quantum dynamical coherence elements, χ_m≠n, a different strategy is needed. As the input state we take a nonmaximally entangled state: |Φ_α⁺⟩ = α|00⟩ + β|11⟩, with

| α |, | β | \notin {0, 1 / \sqrt{2}}

and Im(αβ*) ≠ 0. The sole stabilizer of this state is Z^AZ^B. It is simple to verify that by measuring Z^AZ^B on the output state

E (ρ)

, with ρ = |Φ_α⁺⟩⟨Φ_α⁺|, we obtain the following probabilities for ±1 outcomes:

\begin{matrix} Tr [P_{+ 1} E (ρ)] = χ_{00} + χ_{33} + 2 Re (χ_{03}) ⟨ Z^{A} ⟩ \\ Tr [P_{- 1} E (ρ)] = χ_{11} + χ_{22} + 2 Im (χ_{12}) ⟨ Z^{A} ⟩ \end{matrix}

(32)

where P₊₁ = P_Φ⁺ + P_Φ^–, P_–1 = P_Ψ⁺ + P_Ψ^– and ⟨Z^A⟩ = Tr(ρZ^A) ≠ 0 because of our choice of a nonmaximally entangled input state. The experimental data,

Tr [P_{+ 1} E (ρ)]

and

Tr [P_{- 1} E (ρ)]

, are exactly the probabilities of no bit-flip error and a bit-flip error on qubit A, respectively. Since we already know the {χ_mm} from the population measurement, we can determine Re(χ₀₃) and Im(χ₁₂). After measuring Z^AZ^B the system is in either of the states

ρ_{\pm 1} = P_{\pm 1} E (ρ) P_{\pm 1} / Tr [P_{\pm 1} E (ρ)]

. Next we measure the expectation value of an element of the normalizer

N (S)

of the stabilizer group

S

N (S) = {g \in P_{n} : [S, g] = 0}

(the normalizer preserves the stabilizer under conjugation). For example, we can measure N = X^AX^B, and obtain the measurement results

\begin{matrix} Tr [N ρ_{+ 1}] = \frac{(χ_{00} - χ_{33}) ⟨ N ⟩ + 2 i Im (χ_{03}) ⟨ Z^{A} N ⟩}{Tr [P_{+ 1} E (ρ)]} \\ Tr [N ρ_{- 1}] = \frac{(χ_{11} - χ_{22}) ⟨ N ⟩ - 2 i Re (χ_{12}) ⟨ Z^{A} N ⟩}{Tr [P_{- 1} E (ρ)]}, \end{matrix}

(33)

where ⟨Z^A⟩, ⟨N⟩, and ⟨Z^AN⟩ are all nonzero and already known. eqs 32 and 33 provide a linear system that can be straightforwardly inverted to extract the four independent real parameters needed to calculate the coherence elements χ₀₃ and χ₁₂. It is simple to verify that a simultaneous measurement of the stabilizer and normalizer elements Z^AZ^B and X^AX^B, is again a BSM.

In order to complete the measurement of the remaining coherence elements, we perform an appropriate basis change by preparing the input states H^AH^B|Φ_α⁺⟩ and S^AS^BH^AH^B|Φ_α⁺⟩, which are the eigenvectors of the stabilizer operators X^AX^B and Y^AY^B. Here H and S are single-qubit Hadamard and phase gates acting on the systems A and B. At the output, we measure the stabilizers and a corresponding normalizer element, e.g., Z^AZ^B, which are again equivalent to a standard BSM, and can be expressed by measuring the Hermitian operators P_Φ⁺ ± P_Ψ⁺ and P_Φ^– ± P_Ψ^– (for the input state H^AH^B|Φ_α⁺⟩), and P_Φ⁺ ± P_Ψ^– and P_Φ^– ± P_Ψ⁺ (for the input state S^AS^BH^AH^B|Φ_α⁺⟩). Figure 1(c) illustrates the DCQD scheme.

Overall, in DCQD we only need a single fixed measurement apparatus capable of performing a Bell state measurement, for a complete characterization of the dynamics. This measurement apparatus is used in four ensemble measurements each corresponding to a different input state. Figure 1(c) and Table 1 summarize the preparations required for DCQD in the single qubit case. This table implies that the required resources in DCQD are as follows: (a) preparation of a maximally entangled state (for population characterization), (b) preparation of three other known (nonmaximally) entangled states (for coherence characterization), and (c) a fixed Bell-state analyzer.

Table 1. Input States and Measurements for Direct Characterization of Quantum Dynamics (DCQD) for a Single Qubit^a

	Measurement			Output
Input state	Stabilizer	Normalizer	Bell state measurement	χ_mn
\|Φ⁺⟩	Z^AZ^B, X^AX^B	N/A	P_Ψ^±, P_Φ^±	χ₀₀, χ₁₁, χ₂₂, χ₃₃
\|Φ_α⁺⟩	Z^AZ^B	X^AX^B	P_Φ⁺ ± P_Φ^–, P_Ψ⁺ ± P_Ψ^–	χ₀₃, χ₁₂
\|Φ_α⁺⟩_X	X^AX^B	Z^AZ^B	P_Φ⁺ ± P_Ψ⁺, P_Φ^– ± P_Ψ^–	χ₀₁, χ₂₃
\|Φ_α⁺⟩_Y	Y^AY^B	Z^AZ^B	P_Φ⁺ ± P_Ψ^–, P_Φ^– ± P_Ψ⁺	χ₀₂, χ₁₃

^{^a}

Here |Φ_α⁺⟩ = α|00⟩ + β|11⟩ (), |Φ_α⁺⟩_X(Y) = α|++⟩_X(Y) + β|−–⟩_X(Y) ( and Im(αβ*) ≠ 0) and {|0⟩, |1⟩}, {|±⟩_X}, {|±⟩_Y} are eigenstates of the Pauli operators Z, X, and Y. The fourth column shows the Bell state measurement (BSM) equivalent of the stabilizer + normalizer measurements. The last column is the output: all elements of the process matrix χ.

We have assumed no SPAM errors. However, this assumption can be relaxed in certain situations, in particular when the imperfections are already known. (125,126)

The DCQD scheme has been generalized to higher-dimensional quantum systems, which includes the multiqubit case. (123) The primary advantage of DCQD over all separable QPT methods (in particular, SQPT and separable AAPT; recall that nonseparable AAPT requires many-body interactions, which are experimentally unrealizable) is that the minimal number of required experimental configurations is reduced quadratically from D⁴ⁿ to D²ⁿ, for n qudits of dimension D each. For example, the number of experimental configurations required for systems of three or four physical qubits is reduced from ∼ 5 × 10³ and ∼ 6.5 × 10⁴ in SQPT to 64 and 256 in DCQD, respectively. (26) Moreover, only a single ensemble measurement is required for a commuting set of observables, as was recently rediscovered in the context of entanglement-enhanced learning scheme of Pauli channels. (99)

3.3. Gate Set Tomography

The main disadvantage of QPT is its sensitivity to SPAM errors. Building on quantum state and process tomography, gate set tomography (GST) was introduced for estimation of gate performance in quantum processors while avoiding the impact of SPAM errors, specifically calibration of gate errors and measurements. (31−33) Since its introduction, there have been many experimental implementations that have assessed gate errors by utilizing GST. (128−137)

Unlike quantum process tomography, GST estimates the performance of a gate set without relying on prior calibration of a system, making GST a more generic method for gate performance characterization. Like QPT, GST differs from RB in that it provides a complete characterization of gates, whereas RB provides a single parameter. As an example, RB is insensitive to coherent errors, whereas GST attempts to amplify the effect of coherent gate errors such as rotation and phase errors. (138)

GST is used for estimating the performance of the reconstructed gates after applying the gates defined in a gate set, as we describe below. To start the derivation of the GST method, we need to consider the probability of measuring the system along a certain axis using a measurement operator M_i from a POVM (see Section 2.1) performed after an arbitrary single gate G is applied to a system in the state ρ, which can be written as

p_{i} = ⟨ ⟨ M_{i} | G | ρ ⟩ ⟩ = Tr (M_{i} G ρ)

(34)

where |·⟩⟩ (⟨⟨·|) represents the superket (bra) operators in Hilbert-Schmidt space.

The superoperator notation is commonly used to simplify manipulations with quantum maps. (139) Using this formalism density matrices ρ acting on a Hilbert space of dimension D are “vectorized” and denoted as |ρ⟩⟩, a vector in a Hilbert-Schmidt space of dimension D². The superket |ρ⟩⟩ is a vector created from the matrix representation of an operator by consecutively stacking its columns. Similarly, quantum operations are represented as matrices of dimension of D² × D² (in the same basis as |ρ⟩⟩) with the inner product defined as ⟨⟨A|B⟩⟩ = Tr(A^†B), where A and B can be density operators.

Assuming a processor can perform N_ρ state preparation gates and N_G distinct gates with N_M measurement operators, where the i’th measurement can result in N_M⁽ⁱ⁾ distinct outcomes, we can write down the gate set of the system as

G = {{| ρ^{(j)} ⟩ ⟩}_{j = 1}^{N_{ρ}}; {G_{j}}_{j = 1}^{N_{G}}; ⟨ ⟨ M_{j}^{(m)} |_{m = 1, j = 1}^{N_{M}^{(j)}, N_{M}}}

(35)

This gate set consists of all possible operations that one can perform on the system, where all the gates are dependent on each other. eq 34 implies that the measurement probability for any combination of measurement, gate, and initial state is invariant under a similarity transformation S, i.e.,

\begin{matrix} ⟨ ⟨ M_{j} | \to ⟨ ⟨ M_{j} | S^{- 1} \\ | ρ^{(j)} ⟩ ⟩ \to S | ρ^{(j)} ⟩ ⟩ \\ G \to S G S^{- 1} \end{matrix}

(36)

The outcome of a gate set

G

remains invariant up to an arbitrary invertible S, which suggests choosing a “gauge” S that results in the highest fidelity after reconstructing the gate set.

To make GST independent of the prior calibration of the system and make it robust against SPAM errors, we need to introduce fiducial gates and rewrite the state preparation and measurement gates in eq 35 as

\begin{matrix} | ρ^{(j)} ⟩ ⟩ \to F_{j} | ρ^{(j)} ⟩ ⟩ \\ ⟨ ⟨ M_{i} | \to ⟨ ⟨ M_{i} | F_{i} \end{matrix}

(37)

where F_i is the fiducial circuit constructed using the gates {G_j} in the gate set. Figure 2 shows the circuit required to perform GST on a system.

Figure 2. Gate set tomography circuit consisting of the fiducial gates F_i (F_j) as well as the measurement operators M and the native gate G_k performed on the initial state ρ. Figure adapted from ref (141). Copyright 2021 Quantum.

The most important caveat here is that one needs to make sure that these fiducial circuits form an informationally complete set of state preparation and measurement operators, i.e., measurements that allow for a complete reconstruction of the quantum state. (140)

As an example, we can form a gate set such that

{G} = {I, \sqrt{X}, \sqrt{Y}} = {G_{0}, G_{1}, G_{2}}

with fiducials {F} =

{I, \sqrt{X}, \sqrt{Y}, \sqrt{X} ◦ \sqrt{X}}

, where

\sqrt{X}

and

\sqrt{Y}

represent a π/2 rotation about the Bloch sphere x and y axis, respectively. The choice of

\sqrt{X} ◦ \sqrt{X} = X

instead of X is made here to reduce the experimental cost of implementing the protocol. The key point in this example is that we can span the Bloch sphere (informational completeness) after applying the fiducial gates to the states and measurement operators.

In the final step of GST, we need to define and optimize a likelihood function such that it reproduces the desired gates based on the experimental data. To this end, we first define the outcome probability of applying the GST protocol on the system of interest as (97)

p_{i j k} (\vec{θ}) = ⟨ ⟨ M_{i} (\vec{θ}) | F_{i} (\vec{θ}) G_{k} (\vec{θ}) F_{j} (\vec{θ}) | ρ (\vec{θ}) ⟩ ⟩

(38)

where we combined eqs 34 and 37. Here θ⃗ is a vector consisting of a set of parameters that need to be optimized using the maximum likelihood estimation (MLE) method, i.e., the elements of the T matrix from the Cholesky decomposition introduced below.

The likelihood function can be simplified and written in a normalized form as

L (G) = \sum_{i j k} {(m_{i j k} - p_{i j k} (\vec{θ}))}^{2} / σ_{i j k}^{2}

(39)

with m_ijk representing the measured values in the experiment corresponding to the probabilities p_ijk, and σ_ijk² is the variance in the measurement outcome m_ijk. The remaining task is to minimize the likelihood function [eq 39] using the process matrix χ.

We can write eq 3 in terms of the process matrix χ_G for a gate superoperator

G

G (ρ) = \sum_{i, j = 0}^{D^{2} - 1} {(χ_{G})}_{i j} E_{i} ρ E_{j}

(40)

where {E_i} is a Hermitian operator basis for

B (H)

. Since χ_G is positive semidefinite we can perform a Cholesky decomposition and rewrite it as χ_G = T^†T where T is a lower triangular matrix composed of complex elements on the off-diagonal and real elements on the diagonal. (142)

The trace-preservation property of the density matrix

Tr (G (ρ)) = Tr (ρ)

leads to a completeness relation that can be used to reduce the number of free parameters in χ_G through D² linear equalities:

\sum_{i, j = 0}^{D^{2} - 1} χ_{i j} Tr (E_{i} E_{k} E_{j}) = δ_{k 0}, k = 0, ..., D^{2} - 1

(41)

As mentioned above, θ⃗ is composed of the matrix elements of T, now seen to be (D⁴ – D²)/2.

Rewriting the likelihood function in terms of the process matrices using eq 40 we obtain

p_{i j k} (\vec{θ}) = ⟨ ⟨ M_{i} (\vec{θ}) | F_{i} (\vec{θ}) G_{k} (\vec{θ}) F_{j} (\vec{θ}) | ρ (\vec{θ}) ⟩ ⟩ = Tr [M_{i} F_{i} {G_{k} (F_{j} ρ)}] = \sum_{m n r s t u} {(χ_{F_{j}})}_{t u} {(χ_{G_{k}})}_{r s} {(χ_{F_{i}})}_{m n} \times Tr (M_{i} E_{t} E_{r} E_{m} ρ E_{n} E_{s} E_{u})

(42)

By substituting this expression into eq 39, observing that since M_i is a POVM element we have I – M_i ≥ 0, along with the completeness relation [eq 41] and the unit-trace condition of the density matrix Tr(ρ) = 1, one can solve the MLE problem by varying the elements of the vector θ⃗ (elements of the T matrix). Note that as should be clear from the example we discussed above, the process matrices χ_G and χ_F are related and need to be defined in terms of each other. Moreover, the measurement M_i and state ρ also need to be parametrized similarly to χ_G in terms of their Cholesky decompositions. Optimization of the parameters of θ⃗ after minimizing the likelihood function [eq 39], allows us to reconstruct the entire gate set as well as the process matrices χ_G and χ_F using the Cholesky decomposition mentioned above.

In summary, the following steps are needed to implement the GST protocol:

1.	Initialize the qubit in a pure state: ρ = \|0⟩⟨0\|.
2.	For a particular choice of i, j, and k apply the gate sequence F_iG_kF_j.
3.	Measure the POVM M_i. Typically in experiments, M_i = \|0⟩⟨0\|.
4.	Repeat steps 1–3 many times (typically ∼2000). Average and record the results to obtain m_ijk.
5.	Repeat steps 1–4 for all i, j, and k.
6.	Solve the MLE problem using the likelihood function defined in eq 39 to extract the optimized vector θ⃗ to reconstruct the entire gate set $G$ , as well as the process matrices using the aforementioned Cholesky decomposition.

Another variant of this protocol, known as long-sequence GST, has been developed to amplify and highlight the source of errors. (141) The main difference in this method is that we apply the gates G_k multiple times to amplify the role of coherent errors. The sequence applied is F_i(G_k)^lF_j and l can be varied to estimate the error rates caused by the miscalibration in gates.

In summary, GST provides a comprehensive measure of gate performance in quantum processors. It gives all the details one needs to study and calibrate gates. However, despite recent efforts to simplify the implementation of GST in experiments, (143) GST has major drawbacks such as its inability to capture non-Markovian noise (144) and being extremely resource intensive. For example, to reconstruct the complete gate set for a single qubit one needs to conduct 80 distinct experiments with various gate sequences, with this number increasing to over 4,000 for 2-qubit gates. (97) Note that, as mentioned above, each experiment needs to be averaged over ∼ 2000 runs to minimize the sampling error, making the GST protocol even more resource-intensive. This can make GST impractical in actual experiments, a consequence of the overcompleteness of this method. Recent work has shown that the number of experiments in the two-qubit case can be reduced by a factor of ∼ 8 to elicit only the right amount of information required for gate calibration. (145)

3.4. Direct Fidelity Estimation

Direct fidelity estimation (DFE) (35,36) can be used for estimating how close an implementation of a unitary gate is to the ideal gate, via the average output fidelity metric for quantum maps. That is, DFE provides a single number between 0 and 1 to judge the given gate implementation, where 1 represents the ideal gate implementation. The advantages of this method are that it does not make any assumptions about the underlying noise model and is relatively inexpensive and easy to implement. However, one crucial drawback is that DFE assumes negligible SPAM errors.

DFE essentially provides a Monte Carlo technique to estimate the Uhlmann fidelity between a given and a target pure state. If the given state is

ρ \in B (H)

and the target pure state is

σ \in B (H)

, then the Uhlmann fidelity [recall eq 6] reduces to

F (σ, ρ) = Tr (σ ρ)

(43)

Let {P_k} denote the set of Pauli group operators (including identity) in

B (H)

, normalized so that Tr(P_kP_l) = Dδ_kl, where

D = \dim (H)

. We can then expand

σ = \frac{1}{D} \sum_{k} σ_{k} P_{k}, σ_{k} = Tr (σ P_{k})

(44)

and similarly for ρ. Next, we note that we can recast the fidelity as an expectation over an appropriately defined random variable X that can take values

x_{k} = \frac{ρ_{k}}{σ_{k}}

(45)

with probability

p_{k} = P (X = x_{k}) = \frac{σ_{k}^{2}}{D}

(46)

To see this, note that

\begin{matrix} Tr (σ ρ) & = & \frac{1}{D^{2}} \sum_{k, l} σ_{k} ρ_{l} Tr (P_{k} P_{l}) \\ = & \frac{1}{D} \sum_{k} σ_{k} ρ_{k} = \sum_{k} p_{k} x_{k} = ⟨ X ⟩ \end{matrix}

(47)

Note that eq 46 denotes a normalized probability distribution. The expectation ⟨X⟩ can then be approximated using Monte Carlo techniques, assuming that we are able to sample from the distribution

P

. The rigorous bound on the error of the Monte Carlo procedure is given by the following result (adapted from Theorem 1 of ref (36).):

Theorem 1 Given that

• {i₁, ..., i_N} are N indices sampled from the probability distribution

P

• each Tr(ρP_{i_k}) is estimated using N₂^[k] samples

the fidelity estimate F̅ can be obtained within an error margin of the true fidelity F(σ, ρ) given by

\begin{matrix} P (| F - \bar{F} | < ϵ_{1} + ϵ_{2}) \leq \frac{1}{N ϵ_{1}^{2}} + \\ 2 \exp [- \frac{ϵ_{2}^{2} N^{2}}{2} {(\sum_{k = 1}^{N} \frac{1}{σ_{i_{k}}^{2} N_{2}^{[k]}})}^{- 1}] \end{matrix}

(48)

where σ_i = Tr(σP_i). Here ϵ₁ and ϵ₂ are the errors associated with the Monte Carlo estimate and experimental estimate of Tr(ρP_{i_k}), respectively.

As in Sections 2.1 and 3.2.2, we can invoke the Choi-Jamiołkowski isomorphism (67,68) between quantum processes and quantum states. We can then extend DFE to benchmarking gates. As in eq 4, define the Choi states

ρ_{U} = (I \otimes U) (| ϕ ⟩ ⟨ ϕ |)

and

ρ_{\tilde{U}} = (I \otimes \tilde{U}) (| ϕ ⟩ ⟨ ϕ |)

, where

U

represents the ideal unitary gate operation and

\tilde{U}

represents the actual (noisy) experimental implementation. Then the benchmark metric that is calculable via DFE is the average output fidelity

\bar{F} (U, \tilde{U})

, defined as the average of the process fidelities [eq 6] of

U

and

\tilde{U}

over all possible pure states chosen uniformly at random. It can be shown that (146)

\bar{F} (U, \tilde{U}) = \frac{D F_{proc} (ρ_{U}, ρ_{\tilde{U}}) + 1}{D + 1}

(49)

where the expression on the right is calculable using DFE. A naive implementation would require constructing the maximally entangled state |ϕ⟩. However, one can sidestep the issue of preparing |ϕ⟩ in experiments by simplifying the expression for

F_{proc} (ρ_{U}, ρ_{\tilde{U}})

\begin{matrix} F_{proc} (ρ_{U}, ρ_{\tilde{U}}) & = & \frac{1}{D^{2}} \sum_{j, k} {(ρ_{U})}_{j k} {(ρ_{\tilde{U}})}_{j k} \\ = & \frac{1}{D^{4}} \sum_{j, k} Tr (U (P_{j}^{T}) P_{k}) Tr (\tilde{U} (P_{j}^{T}) P_{k}) \\ = & \frac{1}{D^{4}} \sum_{j, k} Tr (U (P_{j}) P_{k}) Tr (\tilde{U} (P_{j}) P_{k}) \end{matrix}

(50)

where the first line follows from eq 47 with

{(ρ_{E})}_{j k} = Tr (ρ_{E} P_{j} \otimes P_{k})

, the second line due to the identity

Tr [(O \otimes E) (| ϕ ⟩ ⟨ ϕ |)] = \frac{1}{D} Tr (E (O^{T}))

), and the third line from the property of Pauli matrices: P^T = ±P^T. The expression then no longer involves the entangled state |ϕ⟩. The corresponding experimental procedure is schematically illustrated in Figure 3 and becomes

Determine the tuple of Pauli operators (P_j, P_k) for which $Tr (U (P_{j}) P_{k}) \neq 0$ . Call this set A.

Sample a tuple of Pauli operators, each of length log₂(D), from the set A according to the probability distribution $P (P_{j}, P_{k}) = Tr {(U (P_{j}) P_{k})}^{2} / D^{4}$ . This is done via importance sampling. Note that here it is relatively easy to sample because it is possible to calculate the entire probability distribution.

Prepare the D eigenstates for P_j, which require only local operations.

	(a)	For each eigenstate, apply $\tilde{U}$ , and calculate the expectation of P_k (which also requires only local operations and measurements). Call the estimate for the expectation ã_i, and the eigenvalue of the eigenstate λ_i.
	(b)	The estimate for $Tr (\tilde{U} (P_{j}) P_{k})$ is then ∑_iλ_iã_i. Store $\frac{Tr (\tilde{U} (P_{j}) P_{k})}{Tr (U (P_{j}) P_{k})}$ as the estimate for the sample.

Repeat the procedure N times. The average of the sample estimates gives $F (ρ_{U}, ρ_{\tilde{U}})$ .

Figure 3. Schematic of DFE: To estimate the state fidelity, expectations of random Pauli strings P_j are calculated for the input state ρ. The distribution of the random Pauli strings depends on the state with respect to which the fidelity is being calculated.

Alternatively, DFE can also be used for calculating the entanglement fidelity given by

F_{e} = Tr (U^{†} \tilde{U}) / D^{2}

where

U

and

\tilde{U}

are both now treated as matrices acting via left multiplication to give the output state under the corresponding channel. (36)

Theoretically, the number of samples required for DFE scales exponentially with the number of qubits n in the worst case. However, for most states of interest, the number of samples required is only polynomial in n. In practice, DFE is efficient: a variant of it was demonstrated on 7 qubits, (147) while vanilla DFE has been used for a benchmarking implementation of a Toffoli gate using superconducting qubits. (148) However, DFE has been limited in its adaptation to benchmarking due to its susceptibility to SPAM errors; in this sense, alternatives such as RB (Section 3.1) that are robust to SPAM errors are preferred. (53,149,150)

DFE has been extended to simpler and more efficient measurement schemes for fidelity estimation, and tested on multiqubit entangled states on both trapped-ion (151−153) and superconducting quantum devices. (151) Other uses of DFE include tuning control for logic gates in error correcting codes, (154) and calculating minimum gate fidelities. (155)

3.5. Process Fidelity Estimation

An alternative and generalization of DFE is the Process Fidelity Estimation (PFE) method introduced in ref (34). This method is also concerned with estimating the process fidelity between an ideal and actual, noisy process, but generalizes DFE to also account for measurements. As such, it is applicable to entire quantum algorithms.

Consider a projective measurement in the σ_z-basis described by Π(ρ) = ∑_kΠ_k(ρ) = ∑_kπ_kρπ_k (where π_k = |k⟩⟨k| and {π_k} are a complete set of orthogonal projectors), an ideal unitary map

U (ρ) = U ρ U^{†}

, and its noisy implementation

\tilde{U}

. Let the ideal map be

Φ = Π U

and its noisy version be

\tilde{Φ} = Π \tilde{U}

. We can always formally decompose the noisy unitary map as

\tilde{U} = Λ_{U} U

where

Λ_{U} = \tilde{U} U^{†}

is a U-dependent noise map that captures all the noise acting during the noisy implementation of the ideal unitary.

Here, the orthonormal basis states {|i⟩} in the maximally entangled state

| ϕ ⟩ = \frac{1}{\sqrt{D}} \sum_{i = 1}^{D} | i ⟩ \otimes | i ⟩

used to construct the Choi state [eq 4] can be chosen to be the eigenstates of the projectors {π_k}, i.e., π_k|i⟩ = δ_ki|k⟩. Since

Φ U^{†} = Π

and

\tilde{Φ} U^{†} = Π Λ_{U} \equiv \tilde{Π}

, and the Uhlmann fidelity is unitarily invariant, we have

F_{proc} (ρ_{Φ}, ρ_{\tilde{Φ}}) = F_{proc} (ρ_{Π}, ρ_{\tilde{Π}})

(51)

Note that

Π (A) = \sum_{k} π_{k} A π_{k} = \sum_{k} ⟨ k | A | k ⟩ π_{k}

(52)

Writing e_ij ≡ |i⟩⟨j|, the Choi state of Π̃ is given by

ρ_{\tilde{Π}} \equiv (I \otimes \tilde{Π}) (| ϕ ⟩ ⟨ ϕ |) = \frac{1}{D} \sum_{i, j = 1}^{D} e_{i j} \otimes Π [Λ_{U} (e_{i j})] = \frac{1}{D} \sum_{k = 1}^{D} \sum_{i, j = 1}^{D} ⟨ k | Λ_{U} (e_{i j})] | k ⟩ e_{i j} \otimes π_{k} = \frac{1}{D} \sum_{k} \tilde{σ_{k}} \otimes π_{k}

(53)

with

\tilde{σ_{k}} \equiv \sum_{i j} ⟨ k | Λ_{U} (e_{i j}) | k ⟩ e_{i j}

(54)

Similarly, in the noiseless case, i.e., after dropping the tilde everywhere:

ρ_{Π} \equiv (I \otimes Π) (| ϕ ⟩ ⟨ ϕ |) = \frac{1}{D} \sum_{k} σ_{k} \otimes π_{k}

(55)

where:

σ_{k} \equiv \sum_{i, j} ⟨ k | e_{i j} | k ⟩ e_{i j} = \sum_{i, j} δ_{k i} δ_{j k} e_{i j} = π_{k}

(56)

so that

ρ_{Π} = \frac{1}{D} \sum_{k} P_{k}, P_{k} = π_{k} \otimes π_{k}

(57)

and {P_k} is another set of orthogonal projectors: P_kP_l = δ_klP_k. Therefore

ρ_{Π}^{2} = \frac{1}{D^{2}} \sum_{k, l = 1}^{D} P_{k} P_{l} = \frac{1}{D} ρ_{Π}

(58)

so that

\sqrt{D} ρ_{Π}

is the unique positive semidefinite square-root of ρ_Π, i.e.,

\sqrt{ρ} = \sqrt{D} ρ_{Π}

Thus, combining eqs 6 and 51

\begin{matrix} F_{proc} (Φ, \tilde{Φ}) = F_{proc} (Π, \tilde{Π}) = F (ρ_{Π}, ρ_{\tilde{Π}}) \\ = Tr \sqrt{\sqrt{ρ_{Π}} ρ_{\tilde{Π}} \sqrt{ρ_{Π}}} \\ = Tr \sqrt{d ρ_{Π} ρ_{\tilde{Π}} ρ_{Π}} \\ = \frac{1}{D} Tr \sqrt{\sum_{k} a_{k} P_{k}} \end{matrix}

(59)

where a_k = Tr(π_kσ̃_k). Noting that

{(\sum_{k} \sqrt{a_{k}} P_{k})}^{2} = \sum_{k l} \sqrt{a_{k}} \sqrt{a_{l}} P_{k} δ_{k l} = \sum_{k} a_{k} P_{k}

(60)

we can write

F_{proc} (Φ, \tilde{Φ}) = \frac{1}{D} Tr \sum_{k} \sqrt{a_{k}} P_{k} = \frac{1}{D} \sum_{k} \sqrt{a_{k}}

(61)

Now, since

a_{k} = ⟨ k | \tilde{σ_{k}} | k ⟩ = \sum_{i, j} δ_{k i} δ_{j k} ⟨ k | Λ_{U} (e_{i j}) | k ⟩ = ⟨ k | Λ_{U} (π_{k}) | k ⟩

(62)

this finally yields

F_{proc} (Φ, \tilde{Φ}) = \frac{1}{D} \sum_{k = 1}^{D} \sqrt{⟨ k | Λ_{U} (π_{k}) | k} ⟩ = \frac{1}{D} \sum_{k = 1}^{D} \sqrt{\Pr (k | Λ_{U} (π_{k}))}

(63)

where Pr(x|ρ) is the probability of measuring the output x given a state ρ. This result is analogous to the process fidelity expression associated with the DFE method, i.e., eq 50. However, the two methods clearly result in very different operational interpretations.

Since

Λ_{U} (π_{k}) = \tilde{U} [U^{†} (π_{k})]

, eq 63 has the following operational interpretation: it means that we can determine the fidelity of a noisy unitary that is followed by a measurement, by preparing the time-reversed initial states |ψ_k⟩ = U^†|k⟩, applying the noisy unitary

\tilde{U}

and combining the probabilities to measure the respective outputs k. However, this requires the ability to prepare the initial states |ψ_k⟩ with high fidelity, which is a potential drawback of the PFE method. In certain cases this only involves accurately implementable single qubit gates (34) (Hadamard and virtual phase gates).

As the number of different initial states |k⟩ increases exponentially, we can sample m different values {k_l}_l=1...m, where k_l ∈{1, ..., 2ⁿ}, by drawing randomly from the uniform distribution over all 2ⁿ values. Note that using the square of the sample average (denoted by an overline) as an estimator for

E {[\sqrt{X}]}^{2}

has an O(1/m) bias. This can be seen by noting that

E [{(\bar{\sqrt{X}})}^{2}]

E [{(\frac{1}{m} \sum_{i} \sqrt{X_{i}})}^{2}]

E [\frac{1}{m^{2}} (\sum_{i \neq j} \sqrt{X_{i} X_{j}} + \sum_{i} X_{i}]

\frac{m - 1}{m} E {[\sqrt{X}]}^{2}

\frac{1}{m} E [X]

. However, we can construct an unbiased estimator of

E {[\sqrt{X}]}^{2}

by using

\frac{m}{m - 1} {(\bar{\sqrt{X}})}^{2} - \frac{1}{m - 1} \bar{X}

, and estimate the process fidelity via

\begin{matrix} {[F_{proc} (Φ, \tilde{Φ})]}^{2} = {[\frac{1}{D} \sum_{k = 1}^{2^{n}} \sqrt{\Pr (k | Λ_{U} (π_{k}))}]}^{2} \\ \approx \frac{m}{m - 1} {[\frac{1}{m} \sum_{l = 1}^{m} \sqrt{\Pr (k_{l} | Λ_{U} (π_{k}))}]}^{2} \\ - \frac{1}{m (m - 1)} \sum_{l = 1}^{m} \Pr (k_{l} | Λ_{U} (π_{k}))) \end{matrix}

(64)

where l runs over the samples.

This method was used recently in an experimental demonstration of the quantum Fourier transform using midcircuit measurements. (34) Given its novelty, it has not yet been implemented in other experiments.

3.6. Cross-Entropy Benchmarking

Cross-entropy benchmarking (XEB) is a scalable noise characterization and benchmarking scheme that estimates the fidelity F of a n-qubit circuit composed of single- and two-qubit gates. The procedure is based on sampling bitstrings from the output probability distribution of a quantum circuit of interest and was first proposed to characterize the fidelity of running random quantum circuits on a noisy quantum processor, a task known as random circuit sampling (RCS). (37) Later, it was used to produce indirect evidence of quantum supremacy. (39) Following these developments, a new gate characterization method called RCS benchmarking was built on top of XEB and RCS, where the overall noise strength of a circuit is estimated using XEB of random circuits. (156)

Consider the task of sampling bitstrings from the quantum state generated by a circuit. An ideal circuit C_U, which implements a unitary U prepares an ideal pure state ρ_U = |ψ_U⟩⟨ψ_U|, which can be used for sampling. In reality, the bitstrings are drawn from a noisy state ρ_Ũ prepared by the noisy circuit C_Ũ instead. We can model the effect of noise in terms of a simple CPTP map:

ρ_{\tilde{U}} = F_{U} | ψ_{U} ⟩ ⟨ ψ_{U} | + (1 - F_{U}) σ_{U}

(65)

where σ_U denotes an “error state” that captures the effect of the noise, and F_U = ⟨ψ_U|ρ_Ũ|ψ_U⟩ is the fidelity we would like to estimate. Methods such as DFE (see Section 3.4) satisfy precisely this task but have poor worst-case scaling, requiring an exponential number of samples in the number of qubits n. Instead, to compute F, we first define the cross-entropy metric (157) between the ideal and noisy outcome probability distributions:

S (p_{\tilde{U}}, p_{U}) = - \sum_{x \in {0, 1}^{n}} p_{\tilde{U}} (x) \log [p_{U} (x)]

(66)

where p_U(x) = ⟨x|ρ_U|x⟩ and p_Ũ(x) = ⟨x|ρ_Ũ|x⟩ denote the ideal and actual probability distribution of observing the bitstring x from the corresponding ideal and actual states ρ_U and ρ_Ũ respectively. Note that p_U(x) is the probability of bitstring x computed classically for the ideal quantum circuit U, while p_Ũ(x) is obtained from running the actual noisy quantum circuit Ũ using a quantum computer.

When the set of circuits C is random, the expectation of the cross entropy is (38)

E_{U} [S (p_{\tilde{U}}, p_{U})] = - E_{U} [\sum_{x} p_{\tilde{U}} (x) \log [p_{U} (x)]] = E_{U} [F_{U} S (p_{U}, p_{U}) + (1 - F_{U}) S (p_{σ_{U}}, p_{U})]

(67)

where p_{σ_U} = ⟨x|σ_U|x⟩ and we substituted eq 65 in the third line. Under the assumption that the fidelity depends only on the specific gates applied, and not their order, i.e. F_U = F, solving for F yields

F = \frac{E_{U} [S (p_{σ_{U}}, p_{U})] - E_{U} [S (p_{\tilde{U}}, p_{U})]}{E_{U} [S (p_{σ_{U}}, p_{U})] - E_{U} [S (p_{U}, p_{U})]}

(68)

In a loose sense, this reflects an uncorrelated noise model. We further assume that for sufficiently deep circuits, σ_U results in a probability distribution p_{σ_U} which under expectation behaves like a uniform distribution, i.e.,

E_{U} [p_{σ_{U}}]

is a uniform distribution over bitstrings. This is formalized as follows:

\sum_{x} ⟨ x | σ_{U} | x ⟩ \log [p_{U} (x)] = \frac{1}{D} \sum_{x} \log [p_{U} (x)] + ϵ

(69)

where D = 2ⁿ and ϵ decreases sufficiently fast with the number of qubits n. This forms the central assumption in XEB methods. For a depolarizing map, the error state is σ_U = I/D, and eq 69 is exact. More generally, the error ϵ in eq 69 decreases exponentially with n. This can be understood as the “concentration of measure phenomenon” in high-dimensional spaces in the limit of a large number of qubits n. (158)

Eq 69 makes the expression

E_{U} [S (p_{σ_{U}}, p_{U})]

classically calculable. Therefore, in eq 68, only the term

E_{U} [S (p_{\tilde{U}}, p_{U})]

needs to be calculated experimentally; the rest are calculated classically.

In practice, instead of using the logarithm to calculate the standard XEB, a linearized version of XEB is considered. The linear XEB is defined as (39)

S_{linear} (p_{U}, p_{\tilde{U}}) = D \sum_{x \in {0, 1}^{n}} p_{\tilde{U}} (x) p_{U} (x) - 1

(70)

Intuitively, both the standard and linear XEBs are expectations of functions (

E_{X \sim p_{\tilde{U}}} [\log p_{U} (X)]

and

E_{X \sim p_{\tilde{U}}} [p_{U} (X) D - 1]

, respectively) which map observed bitstrings with higher ideal probabilities to larger values. When the circuits are noiseless, in expectation, the bitstring samples are drawn from the ideal probability distribution of random circuits, which is called the Porter-Thomas distribution, (159) an exponential distribution over a random permutation of bitstrings. This is the limit where F = 1. In the opposite limit where the bitstring samples are drawn from the uniform distribution, we have

p_{\tilde{U}} (x) = \frac{1}{D}

and F = 0. The same analysis as eq 68 follows through with the modified, linear XEB definition.

The above exposition describes the theory of using the XEB to calculate the fidelity F. Through heuristic arguments, (38,39,160) the unbiased linear cross entropy is considered an estimator of the fidelity itself.

As in the case of DFE (Section 3.4) and PFE [eq 64], directly computing the exponentially large sum involved in eq 70 is impractical, and instead we resort to a sampling method. To construct an unbiased estimator for the linear cross entropy, consider M bitstring samples S = {x_i}_i=1^M drawn from the probabilities of the noisy output state of a random circuit, the linear cross-entropy fidelity estimator is given by

{\hat{F}}_{XEB} = \frac{D}{M} \sum_{i = 1}^{M} p_{U} (x_{i}) - 1

(71)

Reference (161) proposed the following form for an unbiased linear XEB fidelity estimator:

{\hat{F}}_{uXEB} = \frac{\frac{D}{M} \sum_{i = 1}^{M} p_{U} (x_{i}) - 1}{D \sum_{x \in {0, 1}^{n}} p_{U} {(x)}^{2} - 1}

(72)

Note that one will need to compute all 2ⁿ output probabilities in the denominator of eq 72 to compute the unbiased linear XEB fidelity estimator. For random circuits, the denominator in eq 72 approaches 1 in log depth, (162) hence the two XEB fidelity estimators, eq 71 and eq 72 give the same result as the depth increases. However, the unbiased fidelity estimator F̂_uXEB is more accurate for circuits with short depth.

In both estimators for XEB, {x_i} is sampled from the noisy circuit, while the ideal circuit is classically simulated to provide the values p_U(x_i). A variety of different classical algorithms have been devised for this task, among which tensor networks are the most efficient. (163−165)

Thus, computing F_XEB becomes infeasible for large circuit sizes that do not have tractable classical simulations. This computational hurdle motivates using the XEB measure as a way to certify a quantum computing advantage. More precisely, one can demonstrate a quantum advantage over classical algorithms, for the specific task of estimating the XEB values in the context of the computational task of sampling bitstrings from the probability distributions generated by random circuits. This idea was the basis for the first quantum supremacy experiment. (166)

Among other things, the experimental linear XEB was shown to be consistent with a simple uncorrelated noise model. (166) This fact underlines that sampling from random circuits can be used as a way to benchmark noise in quantum circuits. This is known as RCS benchmarking. (156)

In an RCS benchmarking circuit, similar to the circuit in the IRB protocol (Section 3.1.3), the gate to be characterized is applied in an alternating fashion and interleaved with Haar random single-qubit gates (approximated using unitary t-designs (167,168)), followed by measurements in the computational basis; see Figure 4. The noise introduced by the imperfect gates is assumed to be described by a Pauli noise model of the form

N (ρ) = \sum p_{α} σ_{α} ρ σ_{α}

, where σ_α is an n-qubit Pauli operator and p_α is the associated error probability. The RCS benchmark procedure is the following:

1.	Sample a random circuit C_i from an ensemble of random circuits RCS(n, P) with n qubits and depth P, where P denotes the layers of two-qubit gates independently drawn from the Haar measure on $U (4)$ .
2.	Estimate the fidelity of C_i (denoted F̂_P,i).
3.	Repeat step 1–2 for L instances of random circuits and obtain the average fidelity ${\hat{F}}_{P} = \frac{1}{L} \sum_{i}^{L} {\hat{F}}_{P, i}$ .
4.	Repeat step 1–3 for varying depths $P \in {1, ..., P}$ to obtain a sequence of average fidelities ${{\hat{F}}_{P}}_{1}^{P}$ .
5.	Fit the average fidelities ${{\hat{F}}_{P}}_{1}^{P}$ to an exponentially decaying function F = Ae^–λP, where the fitted parameter λ characterizes the overall noise strength.

Figure 4. RCS benchmarking circuit: layers of 2-qubit gates (white boxes) are interleaved with Haar random 1-qubit gates (blue boxes), followed by measurement in the computational basis at the end. Adapted from Figure 1 of ref (156). Copyright 2021 arXiv.

The key step in the procedure above is estimating the circuit fidelity F, which is defined as the overlap between the output state |ψ_U⟩ of the ideal circuit and the mixed state ρ_Ũ generated by an experiment, i.e., F_U = ⟨ψ_U|ρ_Ũ|ψ_U⟩. This type of fidelity estimate, in general, requires exponentially many measurements to determine the state ρ_Ũ. To circumvent this exponential cost, the cross-entropy fidelity estimator in eq 71 is considered instead.

Under the assumption of a Pauli noise map, it has been shown, both analytically and numerically, (156) that the average fidelities of the random circuits decay exponentially as a function of circuit depth P, i.e.,

E [F] = e^{- λ P}

where λ = ∑_αp_α, which allows one to extract the effective noise rate of the quantum hardware λ by estimating

E [F]

. Similar to the RB protocol, when fitting the experimentally estimated XEB fidelities

{{\hat{F}}_{P}}_{P = 1}^{P}

to an exponentially decaying function, the depth-independent fitting parameters correspond to SPAM errors, which can be estimated independently. For this reason, the XEB benchmark protocol is considered robust to SPAM errors. The exponential decay characteristic and the related extraction of noise parameters can be generalized beyond Pauli noise maps. (164)

Another desirable feature of the XEB benchmark is that the fidelity estimation procedure is sample-efficient. To see this, we consider the variance of the unbiased XEB fidelity estimator, which has been shown to be given by (161)

{Var}_{C, S} ({\hat{F}}_{uXEB}) = O (\frac{1}{M} + λ^{2} {(E [F])}^{2})

(73)

This suggests that the fidelity estimator is sample-efficient (compared to, e.g., DFE) and requires M = O(1/ϵ²) measurement samples to achieve ϵ additive accuracy for each instance of random circuits. The first term in the variance expression is the leading term, especially in large-scale experiments, due to the exponential decay of

E [F]

. In practice, it suffices to collect a large number of samples for a few circuits to estimate

E [F]

with a small additive error.

Unlike the standard RB protocol described in Section 3.1, which involves Clifford gate sequences, the XEB protocol can also be applied to circuits with arbitrary non-Clifford gates. Moreover, RB does not scale easily beyond two or three qubits due to the large circuit depth it requires; XEB, in contrast, is limited only by the classical simulability of the underlying circuit and can extend to up to ∼50 qubits. (156) Hence, it can be used to benchmark broader families of quantum gates and quantum circuits. However, the XEB fidelity estimator can be insensitive to certain types of noise. As an example, under a Pauli noise model, local Pauli errors can add up or cancel as they propagate forward in time through random circuits. When the propagated error operators before the measurement are of type I or Z, they do not contribute to the XEB fidelity as they do not affect measurements in the computational basis. It has also been shown that the XEB is not a good proxy for the fidelity in the presence of time-correlated errors. (169) Overall, the XEB protocol is useful in estimating the total amount of error in a sample-efficient way under certain strong assumptions about the noise. To extract more information about the noise map, alternative noise characterization techniques are needed.

4. Deterministic Benchmarking

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In this section, we introduce a method we call deterministic benchmarking (DB) (170) and validate its utility through experiments conducted on a superconducting transmon qubit. (73) Unlike RB, which averages over many and lengthy random Clifford gate sequences to approximate all errors as a depolarizing map and lacks the sensitivity needed to detect small changes in decoherence times and qubit frequencies, (144,171) DB is highly sensitive to such changes. As illustrated in Figure 5(a,b), DB utilizes a carefully chosen small set of gate-pair sequences that specifically target the real noise affecting single-qubit gates. The DB protocol is significantly more efficient than the other methods we have reviewed in earlier sections but is currently limited to single-qubit gates. Indeed, we show below that DB requires only four sets of state-fidelity measurements to accurately estimate all incoherent and coherent errors relevant to single-qubit gate operations. Additionally, DB can also be used to characterize the bath temperature. Moreover, we show that DB also enables exploration of how coherent and incoherent errors interact to cause biased noise, whose impact on fault tolerance has been thoroughly studied. (83,86)

Figure 5. Schematic representations of the gate sequences for (a) randomized benchmarking (RB) and (b) deterministic benchmarking (DB), respectively. RB uses N random Clifford gates C_i, followed by a single recovery gate C_r = C_N^†···C₁^†. DB prepares an initial state via U|0⟩, applies n repetitions of a fixed, deterministic two-pulse sequence P₁P₂, and unprepares the initial state via U^†.

The gate sequences used in DB resemble the simplest dynamical decoupling (DD) pulse sequences introduced decades ago in nuclear magnetic resonance. (172−174) But, while DD is nowadays typically used to suppress incoherent and coherent noise during idling─whether between gates (34,175−179) or as a result of crosstalk, (180−186) the goal of DB is instead to characterize the noise arising during gate operation.

4.1. Qubit Model

We base our presentation on transmon qubits, (73) which we model as driven two-level systems and consider in the drive frame under the rotating wave approximation. (71) However, the DB methodology is general and applies to any qubit modality.

Let R_α(θ) ≡ exp[−i(θ/2)σ_α], where {σ_α} is the set of Pauli matrices. The Hamiltonian that generates the single-qubit gates R_x(θ) and R_y(θ), including both phase and rotation errors, is given by

H_{α} = (ε + ε_{err}) \frac{σ_{α}}{2} + Δ_{err} \frac{σ_{z}}{2}, α \in {x, y}

(74)

Ideally, both the amplitude error ε_err and the detuning error Δ_err vanish. In reality, both are present and give rise to rotation and phase errors

δ θ \equiv ε_{err} t_{g}, δ ϕ \equiv \frac{Δ_{err}}{\bar{ε}} = \frac{1}{θ} Δ_{err} t_{g}

(75)

respectively, with t_g denoting the gate duration, θ ≡∫₀^t_g ε(t)dt, and ε̅ = θ/t_g the average pulse amplitude. We expect to see a pulse-shape dependence in the phase error δϕ, as the effective detuning error Δ_err may depend on instantaneous drive amplitude ε(t). We assume square-shaped pulses for our analytical theory and cosine-shaped pulses for our numerical calculations using the Lindblad master equation (cosine pulses are used in experimental measurements).

Eq 74 is an effective model, where Δ_err is an error resulting from both detuning─a discrepancy between the drive and qubit frequencies─and the ac-Stark shift induced by driving other transitions (such as the higher levels of the transmon) off-resonantly during the gate. (187) The two-level model is justifiable, given that leakage to higher levels of the transmon during our gates is very small (see Section 4.11).

4.2. Four Key Parameters

Recall from Section 2.1 that any completely positive trace-preserving (CPTP) map acting on a D-dimensional Hilbert space can be characterized using no more than D⁴ – D² real parameters, i.e., 12 for a single qubit. While RB reduces this to a single effective parameter, DB makes a less drastic reduction, to four key parameters: the relaxation time T₁, the coherence time T₂ (which contains information on pure dephasing T_ϕ), the rotation error δθ, and the phase error δϕ. Assuming a Markovian noise model, we show below that these parameters can be straightforwardly extracted and, at the same time, provide just the right amount of information about the errors affecting single-qubit gates to enable such gates to be accurately calibrated.

We denote X ≡ R_x(π), X ≡ R_x(−π), and similarly for Y and Y. To define DB, we need gate-pair sequences of the form P₁P₂. In particular, we consider P₁P₂ ∈{XX, XX, YY, YY, YY}. Note that in the absence of any errors, i.e., when ε_err = Δ_err = 0 and without coupling to a bath, P₁P₂ = ±I for this set (the global phase will not matter). In reality, P₁P₂ ∝̷ I and we consider the corresponding fidelity F̃(n) = |⟨ψ|(P₁P₂)ⁿ|ψ⟩|² where |ψ⟩ is the initial state and

n \in Z^{+}

is the number of repetitions of the P₁P₂ sequence; hence the fidelity is defined stroboscopically. We use a tilde to denote the experimentally measured fidelity.

The core DB protocol consists of a set of four “learning” experiments designed to extract {T₁, T₂, δθ, δϕ}, followed by tests involving new experimental data. In each case, the notation {P₁P₂;|ψ⟩} means a circuit comprising preparation of the state |ψ⟩ from the |0⟩ state, a varying number of repetitions of the P₁P₂ sequence, unpreparation of |ψ⟩, and finally a measurement in the σ^z eigenbasis. Finally, the empirical fidelity F̃ is found for each experiment as the ratio of |0⟩ outcomes to the total number of experimental shots (800 in our case). We find that in all these experiments─learning as well as testing─the fidelity can be accurately described using a simple analytical expression:

F (t_{n}) = \frac{1}{2} (1 + a) + \frac{1}{2} (1 - a) e^{- t_{n} / T_{D}} \cos (2 ω t_{n})

(76)

where t_n = 2nt_g represents the total evolution time. The parameter T_D denotes a characteristic timescale associated with incoherent noise processes, while ω is a frequency associated with coherent errors. The parameter a quantifies the deviation from the infinite temperature limit, where

F (\infty) = \frac{1}{2} .

Eq 76 effectively captures the interplay between coherent and incoherent noise sources. While it is phenomenological, it is also derivable from a Markovian master equation model subject to certain approximations, as shown in Section 4.9.

Finally, we note that eq 76 can be further generalized if we replace

and

, which adds an extra parameter. If we set ω = 0 (no coherent errors), this reduces to an RB fidelity-like expression Apⁿ + B, where p = e^–2t_g/T_D [recall eq 14]. RB’s ability to quantify state preparation and measurement (SPAM) errors as 1 – (A + B) is thus captured by this generalization of DB; however, we do not consider this here, as our focus is the determination of {T₁, T₂, δθ, δϕ}. Finally, note that coherent errors introduced by leakage do not fit the form of eq 76; we discuss this further in Section 4.11.

4.3. Protocol

We now detail the DB protocol, which comprises the following experiments or steps:

1.	{free;\|1⟩}: This is a standard T₁ measurement, where we do not apply any pulses (free evolution). eq 76 reduces to $\frac{1}{2} (1 + a) + \frac{1}{2} (1 - a) e^{- t / T_{1}}$ in this case which we fit to determine T₁. This yields the relaxation component of the Lindbladian.
2.	{XX;\|+⟩}: Fitting this experiment with eq 76 yields T_D = T₂, the coherence time from which we deduce the pure dephasing time T_ϕ = 2T₁T₂/(2T₁ – T₂). This is analogous to an echo-Ramsey measurement (188) and yields the dephasing component of the Lindbladian. This experiment is also sensitive to leakage (discussed below); here we assume leakage is small.
3.	{YY;\|+⟩}: This yields fidelity oscillations which we fit to find the rotation error δθ where ω ≈ δθ/(2t_g) in eq 76.
4.	{XX;\|+⟩}: Similarly, this yields fidelity oscillations which we fit to find the phase error δϕ where ω = δϕ/t_g in eq 76. The decay times T_D obtained here and in step 3 are a combination of the incoherent and coherent noise and hence different from T₂ obtained in step 2. We now have all four parameters, which we use to numerically calculate predictions for the test experiments.
5.	Test 1: {YY;\|+⟩} and {YY;\|+⟩}: This highlights the interplay of asymmetric relaxation noise and gate operations and can also be used to characterize the qubit temperature (see Section 4.10).
6.	Test 2: Various other experiments. Here, we report tests using the UR6 dynamical decoupling sequence. (189)

Next, we demonstrate that this protocol enables a detailed assessment of both coherent and incoherent errors affecting single-qubit gates.

4.4. Experimental Results

We conduct all our experiments using a superconducting transmon qubit dispersively coupled to its readout resonator (see Appendix A for more details). We implement R_y(θ) gates by adjusting the phase of the microwave pulses used for R_x(θ) gates, thus eliminating the need for additional calibrations. All single-qubit gates are calibrated using the Derivative Removal of Adiabatic Gates (DRAG) technique (190) to minimize leakage and phase errors induced by higher levels of the transmon. To emulate drifts in experimental parameters, we first perform measurements on an outdated set of gate calibration parameters. The results are shown in Figure 6.

Figure 6. Empirical (F̃, symbols), analytical fit (F, solid curves), and Lindblad-based (F̅, dashed curves) fidelities for the initial states |1⟩ under free evolution and |+⟩ under various gate sequences P₁P₂. The analytical fit yields T₁ = 23.36 ± 0.40 μs, T₂ = 44.13 ± 2.49 μs, δθ = 0.398 ± 0.004°, and δϕ = 0.426 ± 0.004°. With the exception of the XX (orange) and YY (green) cases, the Lindblad-based results are indistinguishable from the analytical fits.

We observe that F̃_YY oscillates, indicating rotation errors δθ, as explained below. Similarly, F̃_XX oscillates due to phase errors δϕ. The XX sequence on |+⟩ is insensitive to coherent errors and, instead, F̃_XX exhibits an exponential decay characterizing the T₂ time during time-dependent gate operations. The T_D parameter from the XX sequence is chosen as the effective T₂ since, subject to continuous time evolution under the noiseless version of this sequence, starting from |+⟩ the state always remains |+⟩. Also shown in Figure 6 are results from the YY and YY sequences. We discuss the reason for the pronounced difference they exhibit below.

We use eq 76 to fit (solid curves) each of the four experimental fidelities (symbols) in Figure 6 prescribed by steps 1–4 of the DB protocol and find close agreement.

We contrast the resulting {T₁, T₂, δθ, δϕ} values with the single metric r_C derived from the RB protocol [eq 17]. Figure 7 shows the average RB result for 30 distinct experiments with a total length of 700 Clifford gates for the same gate parameters shown in Figure 6. On average, our RB Clifford gate set consists of approximately 80% π/2 and 20% π gates, with π/2 pulses expected to exhibit half the rotation errors and one-fourth of the phase errors. The RB curve can be fitted to a function of the form Ap^m + B (Figure 7). This process yields an average gate infidelity of r_C = (1 – p)/2 = 0.262 ± 0.004% for the single-qubit Clifford gate set, a far less detailed characterization that misses the coherent oscillations and the distinction between T₁ and T₂. This also shows that RB alone is not sufficient to achieve accurate gate calibration. Note that this method is limited by the relaxation time of the qubit (T₁) as well as the gate duration (t_g) with the average single-qubit gate fidelity scaling as ∼ t_g/T₁ (191) and in our case, the gate fidelity is significantly affected by the relatively short T₁ ≈ 20 μs of the device.

Figure 7. Randomized benchmarking measurement for the same gate calibration parameters used in Figure 6. The error bars represent the deviation of the averaged benchmarking results across 30 distinct random sequences with a maximum number of Cliffords (N_Cliffords) of 700, which are smaller than the marker sizes.

As we show in the next several sections, all observed features can be understood using either a simple, analytically solvable closed-system model or a numerical model based on a Lindblad master equation with time-dependent gates. We first provide the analytical fidelities in the closed system case, which exhibit oscillations for certain sequences.

4.5. Fidelity with Coherent Errors

Assuming the system is closed, eq 74 can be used to analytically compute the fidelities of the gate sequences comprising our DB protocol. The closed system fidelity F_P₁P₂^clsd(n) then reflects only the coherent errors contained within P₁ and P₂. For YY and XX and the initial state |+⟩, we find

F_{Y Y}^{clsd} (n) = \cos^{2} (n θ_{err})

(77a)

F_{X X}^{clsd} (n) = 1 - {(\frac{π δ ϕ \sin (n θ_{err})}{θ_{err}})}^{2}

(77b)

where

θ_{err} \equiv \sqrt{{(ε t_{g} + δ θ)}^{2} + {(π δ ϕ)}^{2}}

Expanding to the leading order in δϕ and δθ, with εt_g = π, F_YY^clsd(n) simplifies to cos²(nδθ). Thus, the fidelity under the YY sequence is dominated by rotation errors δθ.

Regarding F_XX^clsd(n), in the small error limit, the second term in eq 77b is suppressed, and F_XX^clsd(n) ≈ 1. Consequently, there is no dependence on either the rotation or the phase error. Finally:

\begin{matrix} F_{X \bar{X}}^{clsd} (n) = \cos^{2} (n φ_{err}) \\ φ_{err} \equiv \tan^{- 1} (2 π \frac{δ ϕ}{θ_{err}} \sin (\frac{θ_{err}}{2}) \frac{\sqrt{1 - Λ / 2}}{1 - Λ}) \\ Λ \equiv {(π \frac{δ ϕ}{θ_{err}})}^{2} (1 - \cos θ_{err}) \end{matrix}

(78)

Expanding to leading order again, with εt_g = π, we have

F_{X \bar{X}}^{clsd} (n) \approx \cos^{2} [2 n δ ϕ (1 - \frac{1}{π} δ θ)]

. Thus, the fidelity under the XX̅ sequence is dominated by phase errors δϕ.

These results justify our assignment of the oscillations observed in the YY and XX experiments shown in Figure 6 to rotation and phase errors, respectively.

4.6. Open System Model

The cosine dependence in eq 76 is well motivated by the closed system (coherent error) fidelities given above, corresponding to T_D = ∞ and a = 0. The choice of the two incoherent parameters in the DB protocol, specifically T₁ (associated with free evolution) and T₂ (associated with the XX sequence), warrants further investigation through numerical simulation to better understand their contributions to the parameters T_D and a in eq 76. We use a simple Lindblad master equation

\partial_{t} ρ = L (ρ)

to capture the effects of nonunitary evolution, with a Lindbladian of the form

L = - i [H, \cdot] + \sum_{α \in {1, ϕ}} γ_{α} (L_{α} \cdot L_{α}^{†} - \frac{1}{2} {L_{α}^{†} L_{α}, \cdot})

(79)

The Lindblad operators are L₁ = σ^– = |0⟩⟨1| and

L_{ϕ} = σ_{z} / \sqrt{2}

with rates γ₁ = 1/T₁ (relaxation) and γ_ϕ = 1/T_ϕ = (2T₁ – T₂)/(2T₁T₂) (dephasing).

Using the fitted parameters {T₁, T₂, δθ, δϕ} from Steps 1–4 of the DB protocol, we numerically solve the Lindblad equation and find good agreement with the experimental results, as shown by the dashed lines in Figure 6. Moreover, knowledge of these parameters allows us to proceed to step 5 of the protocol and predict, as a test, the outcomes of the YY and YY experiments. As shown in Figure 6, there is again good agreement between the simulations and the experimental results. This validates the choice of {T₁, T₂, δθ, δϕ} as an appropriate set of parameters. In contrast, r_C is clearly insufficient for predicting the outcomes of the YY and YY experiments.

4.7. Sensitivity to Coherent Errors

Having confirmed that DB effectively identifies various errors, we next assess the sensitivity of both DB and RB to coherent errors. Initially, we calibrate the gates to minimize both δϕ and δθ (see Appendix B), and immediately follow this calibration by a series of DD experiments, as depicted in Figure 8(a). These experiments focus on the XX and YY sequences, which are sensitive to phase and rotation errors. As a second test (Step 6 in the DB protocol), we include a Universally Robust (URn) DD sequence, specifically designed to suppress coherent errors. (189) We then deliberately introduce coherent errors with (δθ, δϕ) values given in Figure 8. The phase error δϕ, is introduced by mistuning the DRAG weighting parameter α, which effectively varies the detuning error term Δ_err in our model. The slight increase in rotation error δθ from Figure 8(c) (in the absence of δϕ) to (d) (in the presence of δϕ) can be attributed to the second order dependence of θ_err on δϕ, given by

\frac{1}{2} {(π δ ϕ)}^{2} / (π + δ θ)

Figure 8. Experimental fidelities F̃^|+⟩ showing the very different sensitivity of DB and RB to coherent errors. (a) δθ = 0°, δϕ = 0°, (b) δθ = 0°, δϕ = 0.893 ± 0.002°, (c) δθ = 0.932 ± 0.007°, δϕ = 0°, (d) δθ = 0.995 ± 0.009°, δϕ = 0.90 ± 0.002°. These angles are obtained from fits using eq 76. Dashed curves are the numerically computed fidelities F^|+⟩ from the open system Lindblad model using the DB parameters above along with the corresponding T₁ and T₂. The UR6 sequence suppresses even large coherent errors.

In each of cases (a)-(d) we also conduct RB experiments with the same values of Δ_err and δθ. More precisely, when we add coherent errors to the RB gates, they are added to both R(π/2) and R(π) in such a way that for R(π/2) the rotation errors are 1/2 that of R(π) and the phase errors for R(π/2) are 1/4 of that of R(π).

As anticipated, immediately following gate calibration using the parameters extracted from the DB protocol, neither XX nor YY causes oscillations, as shown in Figure 8(a), and the coherent error is negligible. However, after deliberately introducing phase errors, significant oscillations appear in F̃_XX̅, as depicted in Figure 8(b), while F̃_YY simply decays exponentially without exhibiting any oscillations. In contrast, introducing a rotation error in Figure 8(c) reveals clear oscillations in F̃_YY but none in F̃_XX. When both errors are introduced, Figure 8(d) displays clear oscillations for both sequences.

The UR6 sequence never exhibits oscillations, confirming that coherent errors are their source. The Lindblad model with parameters obtained via the DB protocol (dashed curves) is again in good agreement with the experimental fidelities, further confirming the utility of DB. In contrast, the error rate derived from RB, r_C , is unchanged when a phase error (δϕ ≈ 0.90°) is introduced and increases only slightly (≈ 0.02%) with a rotation error (δθ ≈ 0.93°). We see that small coherent errors can cause large, rapid oscillations in sequence fidelity even when the RB fidelity is almost unchanged. This illustrates RB’s limitations in detecting specific types of coherent errors.

Finally, note that in the presence of only detuning error Δ_err in eq 74, the Rabi rate increases from ε to

\sqrt{ε^{2} + Δ_{err}^{2}}

. Thus, for a gate time t_g, there would be an additional induced rotation error

{δ θ}_{ind} = (\sqrt{ε^{2} + Δ_{err}^{2}} - ε) t_{g} ≃ Δ_{err}^{2} t_{g} / (2 ε)

(assuming a square pulse), even without a drive amplitude error. Using Δ_err = ε̅δϕ and ε̅ = θ/t_g, this gives

{δ θ}_{ind} ≃ \frac{1}{2} ϕ {δ ϕ}^{2}

(80)

From the data in Figure 8, we have at most δϕ = 0.90° with a nominal rotation angle of θ = π, giving δθ_ind ≈ 0.02° which is negligible. We therefore ignore the induced rotation error in our analysis.

4.8. Interplay of T₁ Asymmetry and Gates

An intriguing observation arising from Figure 6 is the strong asymmetry in the decay of the YY and YY fidelity curves. Transmon qubits operate at frequencies in the 3 – 6 GHz range and temperatures around 10 mK (∼ 0.2 GHz). In this regime, the thermal energy is negligible, and the ground state |0⟩ is approximately the thermal state of the transmon qubit, i.e., excitation from |0⟩ to |1⟩ is negligible and relaxation dominates. Tracing the Bloch sphere trajectories corresponding to different sequences reveals that YY confines the state to the southern (excited state) hemisphere, while YY limits it to the northern (ground state) hemisphere. Therefore, the former is more susceptible to relaxation (T₁) errors, which explains the faster decay and lower saturation fidelity of YY compared to YY. This asymmetry is also well captured by the time-dependent Lindblad model, as shown by the dashed curves in Figure 6.

To demonstrate this effect, we performed additional experiments in the same outdated gate calibration cycle used in Figure 6. We applied the XX sequence to |±i⟩ (eigenstates of σ_y), as well as to

| ψ_{π / 4, 3 π / 4} ⟩

and

| ψ_{3 π / 4, - 3 π / 4} ⟩

, where |ψ_θ,ϕ⟩ = cos(θ/2)|0⟩ + e^iϕ sin(θ/2)|1⟩. We observe a clear asymmetry between these pairs of orthogonal states under XX, as shown in Figure 9(a). We further confirm this asymmetry using YY applied to |±⟩ (eigenstates of σ_x) and {|ψ_π/4,3π/4⟩, |ψ_{3π/4,–3π/4}⟩, finding the same asymmetric decay patterns both experimentally and numerically in Figure 9(b). These results illustrate the dependence of relaxation during time-dependent gates on the initial state and its trajectory under different pulse sequences. Numerical modeling of this asymmetry enables qubit temperature calibration. (see Section 4.10).

Figure 9. Asymmetry in the decay pattern of DB sequences due to the interplay of gates and T₁. Fidelity decay for (a) XX and (b) YY applied to two pairs of orthogonal initial states. Dashed curves represent the Lindblad master equation simulation results with parameters obtained by fitting eq 76.

4.9. Derivations: Fidelities of the YY and XX Sequences

In this section, we present a detailed analytical derivation of the fidelity subject to Markovian decoherence and coherent errors. We aim to derive our phenomenological fidelity expression, eq 76, as well the specific sequence dependence of the decay constant T_D, the coherent oscillation frequency ω, and the SPAM parameters A and B.

We assume a Markovian environment which we model via the Lindblad master equation with time-dependent driving:

\begin{matrix} \dot{ρ} = - i [H (t), ρ] + L_{D} (ρ) \\ L_{D} (ρ) = \sum_{α} γ_{α} (L_{α} ρ L_{α}^{†} - \frac{1}{2} {L_{α}^{†} L_{α}, ρ}) \end{matrix}

(81)

We recognize that this is a phenomenological model and that rigorously derived master equations for time-dependent driving have a different structure, where the Lindblad operators L_α become time-dependent. (192−198) However, this simplified model is analytically solvable and suffices for our purposes.

4.9.1. Background

For analysis purposes, we work in the Bloch vector picture, which substitutes ρ =

\frac{1}{2}

(I + v_xσ_x + v_yσ_y + v_zσ_z), H =

\frac{1}{2}

(h_xσ_x + h_yσ_y + h_zσ_z), where v = (v_x, v_y, v_z) ∈

R^{3}

, h = (h_x, h_y, h_z) ∈

R^{3}

. The dissipator

L_{D}

is replaced by (R, c) with

R_{i j} \equiv \frac{1}{2} Tr (σ_{i} L_{D} (σ_{j}))

and

c_{i} \equiv \frac{1}{2} Tr (σ_{i} L_{D} (I))

. The Lindblad master equation then is v̇ = h × v + Rv + c (see, e.g., ref (199) for derivations of all these results). The corresponding fidelity with the |+⟩ state is given by

F (t) = \frac{1}{2} (1 + v_{x} (t))

(82)

We model our system as a qubit with an energy gap ω₀₁ coupled to a thermal environment at inverse temperature β along with dephasing noise. The combined noise model translates to R = diag(−γ₂, −γ₂, −γ₁) and c = (0, 0, η γ₁) where

γ_{2} = \frac{1}{2} γ_{1} + γ_{ϕ}

(83)

and

η = \frac{1 - \exp (- β ω_{01})}{1 + \exp (- β ω_{01})} \in [0, 1]

(84)

4.9.2. YY Sequence

For a square YY pulse, h(t) = (0, ε_tot, Δ_err) where ε_tot = ε + ε_err. Then the Bloch vector equation becomes v̇ = Gv + c, where

G = (\begin{matrix} - γ_{2} & - Δ_{err} & ε_{tot} \\ Δ_{err} & - γ_{2} & 0 \\ - ε_{tot} & 0 & - γ_{1} \end{matrix})

(85)

We first solve the case where Δ_err = 0 (δϕ = 0). Define

δ G = (\begin{matrix} 0 & - Δ_{err} & 0 \\ Δ_{err} & 0 & 0 \\ 0 & 0 & 0 \end{matrix})

(86)

Exact Solution for Δ_err = 0

The Bloch vector equation for Δ_err = 0 becomes

\dot{v} = (G - δ G) v + c

(87)

In this equation, v_y is decoupled from the other two components and to solve for v_x(t) and v_z(t), requires exponentiation of a 2 × 2 matrix. Given that v(0) = (1, 0, 0), the relevant solutions for the Bloch vector components take the following form:

where

\begin{matrix} v_{\infty} = η \frac{ε_{tot} γ_{1}}{ε_{tot}^{2} + γ_{1} γ_{2}} \\ γ_{*} = \frac{1}{2} (γ_{1} + γ_{2}) \\ ω_{*} = \sqrt{ε_{tot}^{2} - \frac{1}{4} {(γ_{1} - γ_{2})}^{2}} \end{matrix}

(88)

Along with eq 82, eq 88 is close to, but not quite eq 76.

Perturbation Theory for Δ_err = 0

In the regime where ε ≫ ε_err, γ₁, γ₂, the quantities in eq 88 simplify to

v_{\infty} \approx η \frac{γ_{1}}{ε}, ω_{*} \approx ε_{tot} = ε + ε_{err}

(89)

to first order. Moreover, since we are interested in the fidelities after each DD sequence, i.e., at t = 2nt_g (

n \in N)

, sin(2ω_∗nt_g) ≈ sin(2nδθ) and cos(2ω_∗nt_g) ≈ cos(2nδθ). The term corresponding to sin(2ω_∗nt_g) vanishes to first order initially because it is multiplied by another first-order term and later because it is exponentially suppressed. This results in a simplified expression for v_x(2nt_g):

v_{x} (2 n t_{g}) = v_{\infty} + (1 - v_{\infty}) e^{- 2 n t_{g} γ_{*}} \cos (2 n δ θ)

(90)

The fidelity of the |+⟩ state as a function of YY repetitions n, F_δϕ=0(n), is therefore given by

F_{δ ϕ = 0} (t_{n}) = \frac{1}{2} (1 + v_{\infty}) + \frac{1}{2} (1 - v_{\infty}) e^{- γ_{*} t_{n}} \cos (2 n δ θ)

(91)

Eq 91 is identical in form to eq 76, with a = v_∞ ≈ γ₁/ε, T_D = 1/γ_* and 2ωt_g ≈ δθ. Note that the oscillation period of 2nδθ matches that of the fidelity in the closed system case as discussed below eq 77b.

Perturbation Theory for Δ_err ≠ 0

If the solution of the Bloch vector equation given in eq 87 is denoted by v₀(t), then in the case where Δ_err ≠ 0, let the deviation of the actual Bloch vector trajectory v(t) from v₀(t) be given by

ξ (t) \in R^{3}

, i.e., v(t) = v₀(t) + ξ(t). Substituting this form into the Bloch vector equation v̇ = Gv + c gives

\dot{ξ} = G ξ + δ G v_{0}

(92)

The solution for ξ(t) is then given by

ξ (t) = \int_{0}^{t} e^{G (t - s)} δ G v_{0} (s) d s

(93)

Taking the norm of both sides and moving the norm inside the integral, we obtain

∥ ξ (t) ∥ \leq \int_{0}^{t} ∥ e^{G (t - s)} ∥ ∥ e^{G} δ G v_{0} (s) ∥ d s

(94)

Using eq 86, ∥δGv₀(s)∥ simplifies to |Δ_errv_0,x(s)| since v_0,y(s) = 0. From eq 88, ∥v_0,x(s)∥ in turn can be strictly upper bounded by |v_∞| + ke^–γ_∗s for some constant k. A smaller value of k will result in a tighter bound. In the approximation regime discussed above, k can conveniently be chosen as 2.

If G = SDS^–1 where D is the diagonal matrix of G’s eigenvalues λ_i and S is the similarity matrix with the corresponding eigenvectors of G as its columns, then

∥ e^{G (t - s)} ∥ \leq ∥ S ∥ ∥ e^{D (t - s)} ∥ ∥ S^{- 1} ∥ \leq κ e^{- λ (t - s)}

(95)

where

κ = ∥ S ∥ ∥ S^{- 1} ∥

and

λ = - \max_{i} R (λ_{i})

. (200) This gives the inequality

∥ ξ (t) ∥ < {\begin{array}{l} κ | Δ_{err} | (\frac{1 - e^{- λ t}}{λ} | v_{\infty} | + k \frac{e^{- γ_{*} t} - e^{- λ t}}{λ - γ_{*}}) & if λ \neq γ_{*} \\ κ | Δ_{err} | (\frac{1 - e^{- λ t}}{λ} | v_{\infty} | + k t e^{- λ t}) & otherwise \end{array}

(96)

Note that the first term is O(Δ_errv_∞/λ).

From first-order eigenvalue perturbation theory, we have

δ λ_{i} = \frac{w_{i} \cdot δ G v_{i}}{w_{i} \cdot v_{i}}

(97)

where w_i and v_i are the left and right eigenvectors of G in eq 85 with eigenvalues {λ_i} = {γ_∗ ± iω_∗, −γ₂}. Therefore, using eq 86:

\begin{matrix} λ_{1, 2} = (- γ_{*} \pm i ω_{*}) + O (Δ_{err}) \\ λ_{3} = - γ_{2} + O (Δ_{err}) \end{matrix}

(98)

Under the additional assumption Δ_err/γ_1,2 = O(1), the first summand in eq 96 simplifies to O(v_∞) = O(γ₁/ε). Combining this with the fact that the second summand decays exponentially with time, ξ(t) is a small enough quantity such that v(t) can be well approximated by v₀(t). Therefore, the resulting fidelity expression F_YY(n) in a Δ_err ≠ 0 system is still well captured by eq 91. It matches eq 76 with an error |F_YY(n) – F_δϕ=0(n)| <

\frac{1}{2} ∥ ξ (2 n t_{g}) ∥

, with the factor 1/2 due to eq 82. In big-O notation,

| F_{Y Y} (n) - F_{δ ϕ = 0} (n) | = O (γ_{1} / ε) = O (γ_{1} t_{g})

(99)

4.9.3. XX̅ Sequence

We solve for a simplified model in which T_ϕ = 2T₁, that is, γ₁ = γ₂: all the components of the Bloch vector decay uniformly to the steady state (0, 0, η). The resulting Bloch vector equations are

\dot{v} (t) = {\begin{matrix} G_{+} v (t) + c, & if (2 k) t_{g} < t \leq (2 k + 1) t_{g}, k \in Z \\ G_{-} v (t) + c, & if (2 k - 1) t_{g} < t \leq (2 k) t_{g}, k \in Z \end{matrix}

(100)

where c = (0, 0, η γ), γ = γ₁ = γ₂, and the matrices G_± are

G_{\pm} = (\begin{matrix} - γ & - Δ_{err} & 0 \\ Δ_{err} & - γ & \mp ε_{tot} \\ 0 & \pm ε_{tot} & - γ \end{matrix})

(101)

The solutions for the Bloch vector after each XX sequence (which corresponds to a duration of 2t_g) are then given recursively by

v (2 (n + 1) t_{g}) = A v (2 n t_{g}) - ζ

(102)

where

ζ \in R^{3}

, and

\begin{matrix} A & = & e^{G_{-} t_{g}} e^{G_{+} t_{g}} \\ ζ & = & (I - e^{G_{-} t_{g}}) {(G_{-})}^{- 1} c + e^{G_{-} t_{g}} (I - e^{G_{+} t_{g}}) {(G_{+})}^{- 1} c \end{matrix}

(103)

Unraveling the recursion yields

v (2 n t_{g}) = A^{n} v (0) - \sum_{k = 0}^{n - 1} A^{k} ζ

(104)

We are interested in calculating only v_x(2nt_g):

v_{x} (2 n t_{g}) = (1, 0, 0) \cdot A^{n} v (0) - \sum_{k = 0}^{n - 1} (1, 0, 0) \cdot A^{k} ζ

(105)

The required elements of the matrix Aⁿ can be written as

\begin{matrix} {(A^{n})}_{11} & = & \frac{1}{2} (p^{2 n} + q^{2 n}) \\ {(A^{n})}_{12} & = & \frac{- i ω^{'} (1 + e^{i ω' t_{g}})}{2 χ} (p^{2 n} - q^{2 n}) \\ {(A^{n})}_{13} & = & \frac{- ε_{tot} (1 - e^{i ω' t_{g}})}{2 χ} (p^{2 n} - q^{2 n}) \end{matrix}

(106)

where

\begin{matrix} ω^{'} & \equiv & \sqrt{ε_{tot}^{2} + Δ_{err}^{2}} \\ χ & \equiv & \sqrt{4 e^{i ω' t_{g}} ω^{' 2} + {(1 - e^{i ω' t_{g}})}^{2} Δ_{err}^{2}} \\ p & \equiv & \frac{e^{- γ t_{g} - i ω' t_{g}}}{2 ω^{' 2}} ((1 + e^{2 i ω' t_{g}}) Δ_{err}^{2} + 2 e^{i ω' t_{g}} ε_{tot}^{2} + \\ (1 - e^{i ω' t_{g}}) Δ_{err} χ) \\ q & \equiv & \frac{e^{- γ t_{g} - i ω' t_{g}}}{2 ω^{' 2}} ((1 + e^{2 i ω' t_{g}}) Δ_{err}^{2} + 2 e^{i ω' t_{g}} ε_{tot}^{2} - \\ (1 - e^{i ω' t_{g}}) Δ_{err} χ) \end{matrix}

Substituting these into eq 105 gives

v_{x} (2 n t_{g}) = \frac{1}{2} (p^{2 n} + q^{2 n}) + \frac{1 - p^{2 n}}{2 (1 - p)} (ζ_{x} - \frac{i (1 + e^{i ω' t_{g}}) ω^{'}}{χ} ζ_{y} - \frac{(1 - e^{i ω' t_{g}}) ε_{tot}}{χ} ζ_{z}) + \frac{1 - q^{2 n}}{2 (1 - q)} (ζ_{x} + \frac{i (1 + e^{i ω' t_{g}}) ω^{'}}{χ} ζ_{y} + \frac{(1 - e^{i ω' t_{g}}) ε_{tot}}{χ} ζ_{z})

where ζ carries the temperature dependence. Under first order approximations, χ ≈ 2e^iω′t_g/2ω′, ω′ ≈ ε_tot, and

\begin{matrix} p & \approx & e^{- γ t_{g} - i ϑ} \\ q & \approx & e^{- γ t_{g} + i ϑ} \\ ϑ & \equiv & \tan^{- 1} (2 \sin (\frac{ω^{'} t_{g}}{2}) \frac{Δ_{err}}{ω^{'}}) \end{matrix}

(107)

The values of ζ’s components are lengthy expressions and are given in Appendix C so as not to interrupt the flow. Under first-order approximations, ζ_x ≈ 0, (1 + e^iω′t_g)ζ_y/χ ≈ 0 and ζ_z ≈ 0. Substituting these values gives the following final expression:

\begin{matrix} v_{x} (2 n t_{g}) & \approx & \frac{1}{2} (p^{2 n} + q^{2 n}) \\ = & e^{- 2 n γ t_{g}} \cos 2 n ϑ \end{matrix}

(108)

The expression for the fidelity with the |+⟩ state is, therefore

F (t_{n}) = \frac{1}{2} + \frac{1}{2} e^{- γ t_{n}} \cos (2 n ϑ)

This expression matches eq 76 with 2ωt_g = ϑ, T_D = 1/γ and a = 0. Moreover, the oscillation frequency matches the closed system fidelity oscillation frequency [eq 78] up to the first order (noting that θ_err = ω′t_g).

Solving the general case where γ₁ ≠ γ₂ is difficult: we cannot apply perturbative techniques directly to bound the deviation of the Bloch vector trajectory in a {γ₁ ≠ γ₂, Δ_err ≠ 0} environment (where v_x exhibits oscillating exponential decay) from a Bloch vector trajectory in a {γ₁ ≠ γ₂, Δ_err = 0} environment (where v_x exhibits exponential decay). We rely instead on numerical simulations of the Lindblad equation.

4.10. Finite Temperature

We discuss numerical, experimental, and analytical approaches for accounting for finite-temperature (T > 0) effects.

4.10.1. Numerical

In our simulations, we take the zero temperature limit, which fits our experimental data well and is typically valid for transmons. However, it is straightforward to generalize to a nonzero temperature. For zero temperature, we use a Lindblad operator L₁ = σ^– = |0⟩⟨1| with rate γ₁ = 1/T₁. For a nonzero temperature T, we replace this with two operators L₊ = σ^– = |0⟩⟨1| and L_– = σ⁺ = |1⟩⟨0|, with rates γ_– and γ₊, respectively. The rates obey γ_– + γ₊ = γ₁ = 1/T₁ and the detailed balance condition γ₊/γ_– = exp(−ℏω/k_BT), (64) where ω is the qubit frequency. Varying T in our numerical model thus allows us to fit dynamics at any temperature. In particular, relaxation-sensitive sequences, such as YY̅ applied to |+⟩, saturate to a fidelity that strongly depends on the device temperature.

4.10.2. Experimental

The effect of finite temperature can also be seen in our DB protocol. In general, state preparation and measurement (SPAM) errors will cause the initial state fidelity to be less than 1. We can calibrate these errors out by defining the last point in the T₁ sequence (i.e., the thermally equilibrated state) to be a fidelity of 0, and the first point (i.e., the population-inverted thermal state) to be a fidelity of 1. Such a definition accounts for all temperature effects in state preparation and all measurement errors, rescaling all fidelities by the same multiplicative factor, and thus leaving the fitting and analysis unaffected. If we then compare the saturation fidelities of the XX, YY, and YY sequences to the midpoint of the T₁ fidelities, we see that they differ in a way that depends on the temperature, all saturating to a fidelity of 0.5 in the limit of infinite temperature. We can thus use DB to extract the device temperature, although using the XX sequence (with saturation close to the mixed state) will be less sensitive to small temperature variations than using YY and YY, or typical Rabi-based methods. (201) Nevertheless, our original 4-step protocol captures the effect of finite temperature. Note that this technique is insensitive to measurement error, which can be significant and would disrupt temperature measurements based on single-shot state-assignment readout.

4.10.3. Analytical

Given a in eq 76 and square pulses, in fact, the temperature can be computed without simulation of the Lindblad master equation. This can be shown in the following way: suppose that the Bloch vector equation is the same as eq 100, but with G₊ and G_– replaced with G₁ and G₂, respectively. G₁ and G₂ correspond to the two pulses chosen for the DD sequence; e.g., in an XX sequence, G₁ = G₂ = G₊. Note that G₁ and G₂ carry no temperature dependence. Then, similar to the analysis in Section 4.9.3, the Bloch vector after n pulse-pairs is given by

v (2 n t_{g}) = B^{n} v (0) - η \sum_{k = 0}^{n - 1} B^{k} ζ^{'}

(109)

where similar to eq 103,

\begin{matrix} B & = & e^{G_{2} t_{g}} e^{G_{1} t_{g}} \\ ζ^{'} & = & (I - e^{G_{2} t_{g}}) {(G_{2})}^{- 1} c + e^{G_{2} t_{g}} (I - e^{G_{1} t_{g}}) {(G_{1})}^{- 1} c \end{matrix}

and c = (0, 0, γ₁). The fidelity is then

F (n) = \frac{1 + v (0) \cdot B^{n} v (0) - η \sum_{k = 0}^{n - 1} v (0) \cdot B^{k} ζ^{'}}{2}

(110)

where again η is the thermal factor defined in eq 84. For n → ∞, ∥Bⁿ∥→ 0 from eq 95 and

F (\infty) = \frac{1}{2} (1 - η v (0) \cdot {(I - B)}^{- 1} ζ^{'})

(111)

where we used ∑_k=0^∞ B^k = (I – B)⁻¹. Substituting F(∞) = (1 + a)/2 from eq 76 gives

a = η v (0) \cdot {(B - I)}^{- 1} ζ^{'}

(112)

The LHS of eq 112 is measured experimentally, while the inner product in the RHS is computed using the parameters {T₁, T₂, ε_tot, Δ_err}.

Note that a is maximized when η = 1, i.e., at zero temperature [eq 84]. Accordingly, substituting a_T=0 = (v(0)·(B – I)⁻¹ζ′) into eq 112 and rearranging eq 84 gives the temperature as

k_{B} T = ω_{01} {(\ln (\frac{a_{T = 0} + a}{a_{T = 0} - a}))}^{- 1}

(113)

As mentioned above, for transmons ω₀₁ ∈ [3 – 6]GHz and T ∼ 10 mK, so a ≲ a_T=0. However, for devices where finite temperature effects are significant, a < a_T=0; moreover a ≈ 0 for T →∞.

Since the experimental error in calculating a is largely unaffected by the type of pulse sequence (i.e., by the value of a itself), it is advisable to select pulse sequences that have a higher value of |a_T=0| to minimize the fractional uncertainty in temperature T. E.g., a YY sequence which gives a simulated value of |a_YY| would be preferred over a YY sequence which gives |a_YY| if |a_YY| > |a_YY|.

4.11. Leakage

Leakage to a state outside the qubit subspace is a coherent error as long as the leakage state’s coherence is long-lived compared to the gate duration. The coherent rotation to the leakage state will lead to oscillations of the fidelity. These oscillations will, in principle, show up with any pulse sequence, but they are difficult to distinguish from the oscillations due to rotation errors in the YY sequence or phase errors in the XX sequence. For this reason it is best to look at the XX sequence, which has no oscillations in the absence of leakage. In this case the functional form eq 76 no longer holds, as the amplitude of the oscillations and the amplitude of the decay are decoupled from each other. Because leakage is a dynamical effect that depends on the pulse shape, and because the leakage state may influence the readout signal, no simple analytical formula can capture the effect of leakage on fidelity. Qualitatively, however, it is easy to see that more leakage leads to larger and faster fidelity oscillations. For a simple example, see Figure 10. Here we have done a closed-system simulation of a qubit with a leakage state that is −150 MHz detuned from the qubit transition and has a drive matrix element

\sqrt{2}

times larger─i.e., a qutrit similar to a transmon. (202,203) The simulated pulses have cosine envelopes with durations of 10 or 20 ns (orange squares and blue circles, respectively). When prepared in |+⟩ and driven with the XX sequence, fidelity oscillations appear, with amplitude and frequency that increase as the gate time is decreased (gate bandwidth is increased). Note that these are pulses without DRAG (190) and so leakage can be quite significant. Adding DRAG with appropriate parameters can remove these oscillations. In our experiments the pulses were much slower relative to the anharmonicity and DRAG removed any residual leakage, so we ignore leakage in the experimental analysis.

Figure 10. Simulated survival fidelity of the |+⟩ state after repetitions of the XX sequence, where each gate is a cosine pulse length of 20 ns (blue circles) or 10 ns (orange squares). A third state in the simulation represents leakage, with a detuning (anharmonicity) of −150 MHz relative to the qubit transition. Both the amplitude and frequency of oscillations increase as the gate duration is decreased and leakage becomes more significant.

Because the XX sequence on |+⟩ is unaffected by rotation and phase errors, fidelity oscillations in XX provide a direct diagnostic of the amount of leakage. This is particularly useful in the case where the leakage state is unknown and so measurement of the leaked population may not be possible. While no easy analytical solution presents itself, it should be possible to numerically model a gate sequence and fit the measured data to extract parameters of the leakage state.

4.12. DB Summary and Future Research Directions

DB seeks to strike a balance between the amount of information collected about the noise and efficiency of the process to obtain that information. Moreover, DB highlights noise bias, which might benefit the development of compilation strategies (204) provided circuits can be compiled so as to minimize relaxation by steering quantum trajectories away from excited states. The benefits could extend to resource estimation for fault-tolerant quantum algorithms (205,206) and biased noise error correction. (86) The implementation of the DB protocol is straightforward and adaptable beyond superconducting qubits. We anticipate that DB will contribute to the benchmarking and enhancement of quantum gates across various platforms. A natural open problem is to extend DB to two-qubit gates and identify the corresponding minimal extended parameter set that describes incoherent and coherent errors.

5. Conclusions and Outlook

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Gate benchmarking techniques are crucial for practical quantum computing, fulfilling two key roles: (1) offering realistic estimates of attainable performance with existing quantum hardware, and (2) supplying valuable metrics for assessing and monitoring advancements in various gate and circuit implementations. These methods differ widely in their assumptions about the computation and error models, their implementation’s simplicity and cost, and the level of detail they provide about the noise and errors impacting the gates.

This review provides a comprehensive analysis of key benchmarking methodologies for assessing noise in quantum gates and circuits, emphasizing the trade-offs inherent in each strategy. Among the techniques explored are Randomized Benchmarking (RB), well-known for its efficient and scalable computation of average gate fidelity; Quantum Process Tomography, which, while entailing greater implementation costs, yields a complete noise characterization of gates but is sensitive to SPAM; Gate Set Tomography, which has all the ingredients of quantum process tomography but includes state preparation and measurement errors explicitly in its model; Process Fidelity Estimation and Direct Fidelity Estimation, which deliver practical fidelity measures with reduced resource demands and are applicable to both circuits and individual gates; as well as Cross Entropy Benchmarking and Random Circuit Sampling, which are predominantly employed for evaluating complete quantum circuits, particularly in the context of demonstrating quantum supremacy. We also presented a new Deterministic Benchmarking (DB) protocol and demonstrated it using a superconducting transmon qubit. DB is deterministic and efficient: it employs a small and fixed set of simple pulse-pair sequences. The protocol quantifies both incoherent and coherent gate errors that elude RB and does so without averaging over random circuits, in contrast to all the other methods we surveyed, with the exception of QPT.

As quantum computing technology reaches maturity, benchmarking will become a valuable tool for fine-tuning these systems. Currently, however, it is essential for identifying noise and decoherence sources, as well as aiding in the development and optimization of quantum information devices until they achieve reliability and functionality. Ongoing enhancement and advancement of benchmarking methods are pivotal for raising the fidelity of quantum logic gates and circuits to meet the demands of scalable quantum computing architectures.

Ultimately, combining refined noise characterization and error mitigation holds the key to unlocking challenging quantum chemistry problems that extend beyond the scope of classical computation. (21) In the near term, these improvements enable the design of noise-resilient ansätze for VQE, thereby raising the ceiling on the system sizes that can be reliably tackled on noisy intermediate-scale quantum (NISQ) devices. (20) Over the longer horizon, as fault-tolerance thresholds are approached and eventually surpassed, the careful measurement of device-specific noise will continue to inform quantum error correction procedures, bridging the gap between hardware capabilities and the exacting precision required for practical chemical applications. By continuing to improve noise characterization and error mitigation, we inch closer to a regime where quantum hardware can deliver transformative insights into the molecular processes and material properties that govern our physical world.

Author Information

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Corresponding Authors
- Eli M. Levenson-Falk - Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, United States; Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States; Ming Hsieh Department of Electrical & Computer Engineering, University of Southern California, Los Angeles, California 90089, United States; https://orcid.org/0000-0002-0019-9465; Email: [email protected]
- Daniel A. Lidar - Department of Chemistry, University of Southern California, Los Angeles, California 90089, United States; Ming Hsieh Department of Electrical & Computer Engineering, University of Southern California, Los Angeles, California 90089, United States; Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, United States; Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States; https://orcid.org/0000-0002-1671-1515; Email: [email protected]
Authors
- Vinay Tripathi - Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, United States; Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States; https://orcid.org/0000-0003-1418-9046
- Daria Kowsari - Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, United States; Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States
- Kumar Saurav - Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States; Ming Hsieh Department of Electrical & Computer Engineering, University of Southern California, Los Angeles, California 90089, United States
- Haimeng Zhang - Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, United States; Ming Hsieh Department of Electrical & Computer Engineering, University of Southern California, Los Angeles, California 90089, United States
Author Contributions
V.T. and D.K. contributed equally to this work.
Notes
The authors declare no competing financial interest.

Biographies

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Vinay Tripathi received his PhD in Physics from the University of Southern California in 2024, focusing on noise characterization and suppression in superconducting quantum processors. Upon graduation, he joined the IBM Quantum team as a Research Scientist, contributing to advancements in quantum computing hardware and noise mitigation techniques.

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Daria Kowsari received his PhD in Physics from Washington University in St. Louis in 2023. Upon graduation, he joined USC as a postdoctoral scholar. His research focuses on the realization of novel open quantum systems utilizing qubit processors as well as developing new methods to characterize the errors in quantum gates and circuits.

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Kumar Saurav is a PhD student in the ECE department of the University of Southern California and is a Viterbi Graduate Fellowship recipient. Prior to USC, Saurav completed his Bachelor’s degree with Honors in Computer Science from IIT Bombay, following which he worked as a Software Engineer at Google for two years. His current research interests mainly include quantum control and device physics.

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Haimeng Zhang received her PhD in Electrical Engineering from the University of Southern California in 2023. Her research focuses on noise characterization and suppression in superconducting qubits. After graduation, she joined IBM as a quantum algorithm engineer, where she works on benchmarking transpilation methods and developing error suppression and mitigation techniques to improve the implementation of quantum algorithms on quantum computers.

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Eli M. Levenson-Falk is an Associate Professor in the Department of Physics & Astronomy and the Ming Hsieh Department of Electrical & Computer Engineering at the University of Southern California. He is the principal investigator of the Levenson-Falk Lab (LFL) which conducts experimental research using superconducting circuits. Eli received his PhD in Physics from the University of California Berkeley in 2013. He is a 2018 Air Force Office of Scientific Research Young Investigator, a 2021 Office of Naval Research Young Investigator, and a 2021 Cottrell Scholar.

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Daniel A. Lidar is the holder of the Viterbi Professorship of Engineering at the University of Southern California, and a Professor of Electrical & Computer Engineering, Chemistry, and Physics & Astronomy. He holds a PhD in Physics from the Hebrew University of Jerusalem. He did his postdoctoral work in Theoretical Chemistry at UC Berkeley. Prior to joining USC in 2005 he was a faculty member at the University of Toronto. His main research interest is quantum information processing, where he works on quantum control, quantum error correction, the theory of open quantum systems, and quantum algorithms. He is the Director of the USC Center for Quantum Information Science and Technology, the Director of the USC-IBM Quantum Innovation Center, and the co-Director (Scientific Director) of the USC Center for Quantum Computing. He is a recipient of a Sloan Research Fellowship, a Guggenheim Fellowship and is a Fellow of the AAAS, APS, and IEEE.

Acknowledgments

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We thank Namit Anand, Nic Ezzell and Alexandru Paler for useful discussions. Devices were fabricated and provided by the Superconducting Qubits at Lincoln Laboratory (SQUILL) Foundry at MIT Lincoln Laboratory, with funding from the Laboratory for Physical Sciences (LPS) Qubit Collaboratory. This research was supported by the National Science Foundation, the Quantum Leap Big Idea under Grant No. OMA-1936388, the U. S. Army Research Laboratory and the U. S. Army Research Office contract/grant number W911NF2310255, and the Intelligence Advanced Research Projects Activity (IARPA) under Cooperative Agreement No. W911NF-23-2-0216.

Appendix

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This appendix describes technical details relevant to the Deterministic Benchmarking protocol.

A. Device Design and Experimental Setup

The device used in the experiment consists of a simple grounded Xmon geometry (207) dispersively coupled to its co-planar waveguide (CPW) readout resonator. It is nominally identical to the device reported in Ref (95)., and the design is posted on the open-source SQuADDS database. (208) We make use of a separate weakly-coupled microwave line to drive the qubit, which results in reasonably fast gate times (80 ns) with an average gate fidelity of 99.74%. The device layout was created using the Qiskit Metal open-source package (209) with the finite-element (FE) simulations taking place in the Ansys HFSS simulator to find the various parameters of the device, i.e., the qubit and cavity frequencies (ω_q and ω_c), as well as the electric field distribution across the device of interest. The FE simulation results are then analyzed using the pyEPR package (210) to extract the qubit anharmonicity (η) and the qubit-cavity dispersive shift (χ_qc). Lastly, the resonator line width (κ) is extracted using the HFSS driven-modal scattering simulations. Table 2 shows the measured parameters of the device used in this paper.

Table 2. Measured Parameters of the Device Used in the Experiment

ω_q/2π	\|η\|/2π	χ_qc/2π	ω_c/2π
4.37 GHz	215 MHz	230 kHz	6.95 GHz

κ/2π	T₁	T₂^*
150 kHz	20 μs	32 μs

The device is fabricated and packaged in a copper box. We then mount and thermalize the box in the lowest stage of a dilution refrigerator (DR), which has a base temperature of ∼ 15mK, surrounded by an additional copper shielding as well as a Cryoperm to further isolate the chip from the external infrared radiation and magnetic fields. The readout and drive lines each have 50 and 60 dB of attenuation to reach the single-photon power regime on the chip. Moreover, we use an 8 GHz low-pass filter on each line to protect the device from high-frequency noise, as well as high-frequency Eccosorb filters mounted close to the device in the DR to repel the infrared radiation from the higher stages of the DR. The readout line is protected by using four cryogenic microwave circulators, with the signal being amplified using both a traveling wave parametric amplifier (TWPA) at the mixing chamber and a high-electron-mobility transistor (HEMT) amplifier at the 4K stage to achieve single-shot readout.

B. Conventional Single-Qubit Gate Calibration

In our experiments, the H_x Hamiltonian [eq 74] is fully calibrated and the H_y Hamiltonian is then obtained simply by applying a phase shift via an arbitrary waveform generator (AWG), which is equivalent to

H_{y} = R_{z} (- \frac{π}{2}) H_{x} R_{z} (\frac{π}{2})

, and leaving ε_err and Δ_err unchanged.

We use a cosine envelope pulse shape for all the experiments presented in this work. All our pulses are 80 ns long with an additional padding of 4 ns on both sides, i.e., t_g = 88 ns. We perform the following steps to calibrate the single-qubit gates used in the experiments:

1.	Find the resonance frequencies of the cavity and the qubit using conventional spectroscopy techniques.
2.	Look for a monotone signal sent at the frequency of the qubit (cavity) at the output spectrum of the microwave mixers and try to minimize the leakage and the mirror signals caused by the mixer’s electrical offset and imbalance.
3.	Perform simple Rabi experiments to find the right gate length for a specific gate amplitude. We chose a gate length of 80 ns and a cosine-shaped gate envelope for the experiments reported here. Note that there is a padding of 8 ns between consecutive pulses, and hence the effective pulse interval is 88 ns.
4.	Characterize the relaxation and decoherence rates of the device and use the Ramsey experiments to fine-tune the qubit’s frequency.
5.	Optimize the readout fidelity by using the previously calibrated R_x(π) pulses and varying the readout tone’s frequency and amplitude.
6.	Apply consecutive R_x(π) and R_x(π/2) pulses to the system initialized in the ground state, such that after each measurement the qubit ideally returns to its ground state, i.e., 4N × R_x(π/2) and 2N × R_x(π) pulses with N ∈ [1, ..., 20]. Using this method we can observe the ground state fidelity of the qubit and pick an amplitude that pins the qubit to its ground state without exhibiting any oscillations. As described above, calibrating the pulses along the x-axis automatically calibrates them along the y-axis as well.
7.	Optimize the DRAG weighting parameter (α) by applying multiple pairs of R_x(π)R_x(−π) pulses to the system initialized in its ground state, (211) observe the ground state fidelity, and choose the parameter values for which the qubit remains in its ground state.

C. ζ’s Components

The values of ζ’s components [defined in eq 103], along with their first order approximations (Δ_err, γ, ε_err ≪ ε = π/t_g) are noted below:

ζ_{x} = \frac{- 4 η e^{- 2 γ t_{g}} Δ_{err} ε_{tot}}{ω^{' 4} (ω^{' 2} + γ^{2})} [ω^{' 2} ({(γ \sin \frac{ω^{'} t_{g}}{2} + \frac{(1 - e^{γ t_{g}})}{2} ω^{'} \cos \frac{ω^{'} t_{g}}{2})}^{2} + \frac{ω^{' 2}}{4} {((1 - e^{γ t_{g}}) \sin \frac{ω^{'} t_{g}}{2})}^{2}) - Δ_{err}^{2} (1 - \cos (ω^{'} t_{g})) ({(γ \sin \frac{ω^{'} t_{g}}{2} + \frac{1}{2} ω^{'} \cos \frac{ω^{'} t_{g}}{2})}^{2} + \frac{ω^{' 2}}{4} (2 (1 - e^{γ t_{g}}) - \cos^{2} \frac{ω^{'} t_{g}}{2}))] \approx 0

ζ_{y} = \frac{η e^{- 2 γ t_{g}} ω^{'} ε_{tot}}{ω^{' 4} (ω^{' 2} + γ^{2})} [- 4 Δ_{err}^{2} \sin (ω^{'} t_{g}) ({(γ \sin \frac{ω^{'} t_{g}}{2} + \frac{1}{2} ω^{'} \cos \frac{ω^{'} t_{g}}{2})}^{2} + \frac{ω^{' 2}}{4} (2 (1 - e^{γ t_{g}}) - \cos^{2} \frac{ω^{'} t_{g}}{2})) + γ ω^{' 3} ({(e^{γ t_{g}} - \cos (ω^{'} t_{g}))}^{2} + \sin^{2} (ω^{'} t_{g}))] \approx {(1 + e^{- γ t_{g}})}^{2} \frac{γ}{ε}

ζ_{z} = \frac{η e^{- 2 γ t_{g}}}{ω^{' 4} (ω^{' 2} + γ^{2})} [8 Δ_{err}^{2} ε_{tot}^{2} \sin^{2} \frac{ω^{'} t_{g}}{2} ({(γ \sin \frac{ω^{'} t_{g}}{2} + \frac{1}{2} ω^{'} \cos \frac{ω^{'} t_{g}}{2})}^{2} - \frac{ω^{' 2}}{4} \cos^{2} \frac{ω^{'} t_{g}}{2}) + e^{γ t_{g}} γ ω^{' 3} (e^{γ t_{g}} γ ω^{'} + 2 ε_{tot}^{2} \sin (ω^{'} t_{g})) + {Δ_{err}}^{2} ω^{' 2} (e^{γ t_{g}} - 1) (e^{γ t_{g}} ω^{' 2} + (2 \cos (ω^{'} t_{g}) - 1) ε_{tot}^{2} + {Δ_{err}}^{2})] \approx 0

References

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This article references 211 other publications.

1
Bharti, K.; Cervera-Lierta, A.; Kyaw, T. H.; Haug, T.; Alperin-Lea, S.; Anand, A.; Degroote, M.; Heimonen, H.; Kottmann, J. S.; Menke, T.; Mok, W.-K.; Sim, S.; Kwek, L.-C.; Aspuru-Guzik, A. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 2022, 94, 015004 DOI: 10.1103/RevModPhys.94.015004
Google Scholar
There is no corresponding record for this reference.
2
Dalzell, A. M.; McArdle, S.; Berta, M.; Bienias, P.; Chen, C.-F.; Gilyén, A.; Hann, C. T.; Kastoryano, M. J.; Khabiboulline, E. T.; Kubica, A.; Salton, G.; Wang, S.; Brandão, F. G. S. L. Quantum algorithms: A survey of applications and end-to-end complexities. 2023; https://arxiv.org/abs/2310.03011 (accessed 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
3
Shor, P. Fault-tolerant quantum computation. Proceedings of 37th Conference on Foundations of Computer Science. 1996; pp 56– 65.
Google Scholar
There is no corresponding record for this reference.
4
Aharonov, D.; Ben-Or, M. Fault tolerant quantum computation with constant error. Proceedings of 29th Annual ACM Symposium on Theory of Computing (STOC). New York, NY, 1997; p 176.
Google Scholar
There is no corresponding record for this reference.
5
Knill, E.; Laflamme, R.; Zurek, W. H. Resilient Quantum Computation. Science 1998, 279, 342– 345, DOI: 10.1126/science.279.5349.342
Google Scholar
There is no corresponding record for this reference.
6
Aliferis, P.; Gottesman, D.; Preskill, J. Quantum accuracy threshold for concatenated distance-3 codes. Quant. Inf. Comput. 2006, 6, 97, DOI: 10.26421/QIC6.2-1
Google Scholar
There is no corresponding record for this reference.
7
Campbell, E. T.; Terhal, B. M.; Vuillot, C. Roads towards fault-tolerant universal quantum computation. Nature 2017, 549, 172, DOI: 10.1038/nature23460
Google Scholar
There is no corresponding record for this reference.
8
Reiher, M.; Wiebe, N.; Svore, K. M.; Wecker, D.; Troyer, M. Elucidating reaction mechanisms on quantum computers. Proc. Natl. Acad. Sci. U. S. A. 2017, 114, 7555– 7560, DOI: 10.1073/pnas.1619152114
Google Scholar
There is no corresponding record for this reference.
9
Kassal, I.; Whitfield, J. D.; Perdomo-Ortiz, A.; Yung, M.-H.; Aspuru-Guzik, A. Simulating Chemistry Using Quantum Computers. Annu. Rev. Phys. Chem. 2011, 62, 185– 207, DOI: 10.1146/annurev-physchem-032210-103512
Google Scholar
There is no corresponding record for this reference.
10
Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2018, 2, 79, DOI: 10.22331/q-2018-08-06-79
Google Scholar
There is no corresponding record for this reference.
11
O’Malley, P. J. J. Scalable Quantum Simulation of Molecular Energies. Physical Review X 2016, 6, 031007 DOI: 10.1103/PhysRevX.6.031007
Google Scholar
There is no corresponding record for this reference.
12
Kandala, A.; Mezzacapo, A.; Temme, K.; Takita, M.; Brink, M.; Chow, J. M.; Gambetta, J. M. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 2017, 549, 242, DOI: 10.1038/nature23879
Google Scholar
There is no corresponding record for this reference.
13
Lidar, D. A.; Wang, H. Calculating the thermal rate constant with exponential speedup on a quantum computer. Phys. Rev. E 1999, 59, 2429– 2438, DOI: 10.1103/PhysRevE.59.2429
Google Scholar
There is no corresponding record for this reference.
14
Aspuru-Guzik, A.; Dutoi, A. D.; Love, P. J.; Head-Gordon, M. Simulated Quantum Computation of Molecular Energies. Science 2005, 309, 1704, DOI: 10.1126/science.1113479
Google Scholar
There is no corresponding record for this reference.
15
Peruzzo, A.; McClean, J.; Shadbolt, P.; Yung, M.-H.; Zhou, X.-Q.; Love, P. J.; Aspuru-Guzik, A.; O’Brien, J. L. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 2014, 5, 4213, DOI: 10.1038/ncomms5213
Google Scholar
There is no corresponding record for this reference.
16
McClean, J. R.; Romero, J.; Babbush, R.; Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 2016, 18, 023023 DOI: 10.1088/1367-2630/18/2/023023
Google Scholar
There is no corresponding record for this reference.
17
McArdle, S.; Endo, S.; Aspuru-Guzik, A.; Benjamin, S. C.; Yuan, X. Quantum computational chemistry. Rev. Mod. Phys. 2020, 92, 015003 DOI: 10.1103/RevModPhys.92.015003
Google Scholar
There is no corresponding record for this reference.
18
Tilly, J.; Chen, H.; Cao, S.; Picozzi, D.; Setia, K.; Li, Y.; Grant, E.; Wossnig, L.; Rungger, I.; Booth, G. H.; Tennyson, J. The Variational Quantum Eigensolver: A review of methods and best practices. Phys. Rep. 2022, 986, 1– 128, DOI: 10.1016/j.physrep.2022.08.003
Google Scholar
There is no corresponding record for this reference.
19
Kitaev, A. Y. Quantum measurements and the Abelian Stabilizer Problem. 1995; https://arxiv.org/abs/quant-ph/9511026 (accessed 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
20
Cao, Y.; Romero, J.; Olson, J. P.; Degroote, M.; Johnson, P. D.; Kieferová, M.; Kivlichan, I. D.; Menke, T.; Peropadre, B.; Sawaya, N. P. D.; Sim, S.; Veis, L.; Aspuru-Guzik, A. Quantum Chemistry in the Age of Quantum Computing. Chem. Rev. 2019, 119, 10856– 10915, DOI: 10.1021/acs.chemrev.8b00803
Google Scholar
There is no corresponding record for this reference.
21
Lee, S. Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry. Nat. Commun. 2023, 14, 1952, DOI: 10.1038/s41467-023-37587-6
Google Scholar
There is no corresponding record for this reference.
22
Huggins, W. J.; McClean, J. R.; Rubin, N. C.; Jiang, Z.; Wiebe, N.; Whaley, K. B.; Babbush, R. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. npj Quantum Information 2021, 7, 23, DOI: 10.1038/s41534-020-00341-7
Google Scholar
There is no corresponding record for this reference.
23
Hashim, A. . A Practical Introduction to Benchmarking and Characterization of Quantum Computers. 2024; https://arxiv.org/abs/2408.12064 (accessed 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
24
Chuang, I. L.; Nielsen, M. A. Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt. 1997, 44, 2455, DOI: 10.1080/09500349708231894
Google Scholar
There is no corresponding record for this reference.
25
Poyatos, J. F.; Cirac, J. I.; Zoller, P. Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate. Phys. Rev. Lett. 1997, 78, 390– 393, DOI: 10.1103/PhysRevLett.78.390
Google Scholar
There is no corresponding record for this reference.
26
Mohseni, M.; Rezakhani, A. T.; Lidar, D. A. Quantum-process tomography: Resource analysis of different strategies. Phys. Rev. A 2008, 77, 032322 DOI: 10.1103/PhysRevA.77.032322
Google Scholar
There is no corresponding record for this reference.
27
Emerson, J.; Alicki, R.; Życzkowski, K. Scalable noise estimation with random unitary operators. Journal of Optics B: Quantum and Semiclassical Optics 2005, 7, S347, DOI: 10.1088/1464-4266/7/10/021
Google Scholar
There is no corresponding record for this reference.
28
Knill, E.; Leibfried, D.; Reichle, R.; Britton, J.; Blakestad, R. B.; Jost, J. D.; Langer, C.; Ozeri, R.; Seidelin, S.; Wineland, D. J. Randomized benchmarking of quantum gates. Phys. Rev. A 2008, 77, 012307 DOI: 10.1103/PhysRevA.77.012307
Google Scholar
There is no corresponding record for this reference.
29
Magesan, E.; Gambetta, J. M.; Emerson, J. Scalable and Robust Randomized Benchmarking of Quantum Processes. Phys. Rev. Lett. 2011, 106, 180504, DOI: 10.1103/PhysRevLett.106.180504
Google Scholar
There is no corresponding record for this reference.
30
Helsen, J.; Roth, I.; Onorati, E.; Werner, A. H.; Eisert, J. General Framework for Randomized Benchmarking. PRX Quantum 2022, 3, 020357 DOI: 10.1103/PRXQuantum.3.020357
Google Scholar
There is no corresponding record for this reference.
31
Merkel, S. T.; Gambetta, J. M.; Smolin, J. A.; Poletto, S.; Córcoles, A. D.; Johnson, B. R.; Ryan, C. A.; Steffen, M. Self-consistent quantum process tomography. Phys. Rev. A 2013, 87, 062119 DOI: 10.1103/PhysRevA.87.062119
Google Scholar
There is no corresponding record for this reference.
32
Blume-Kohout, R.; Gamble, J. K.; Nielsen, E.; Mizrahi, J.; Sterk, J. D.; Maunz, P. Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit. 2013; https://arxiv.org/abs/1310.4492 (accessed 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
33
Kim, D.; Ward, D. R.; Simmons, C. B.; Gamble, J. K.; Blume-Kohout, R.; Nielsen, E.; Savage, D. E.; Lagally, M. G.; Friesen, M.; Coppersmith, S. N.; Eriksson, M. A. Microwave-driven coherent operation of a semiconductor quantum dot charge qubit. Nat. Nanotechnol. 2015, 10, 243– 247, DOI: 10.1038/nnano.2014.336
Google Scholar
There is no corresponding record for this reference.
34
Bäumer, E.; Tripathi, V.; Seif, A.; Lidar, D.; Wang, D. S. Quantum Fourier Transform Using Dynamic Circuits. Phys. Rev. Lett. 2024, 133, 150602– 150609, DOI: 10.1103/PhysRevLett.133.150602
Google Scholar
There is no corresponding record for this reference.
35
da Silva, M. P.; Landon-Cardinal, O.; Poulin, D. Practical characterization of quantum devices without tomography. Phys. Rev. Lett. 2011, 107, 210404– 210409, DOI: 10.1103/PhysRevLett.107.210404
Google Scholar
There is no corresponding record for this reference.
36
Flammia, S. T.; Liu, Y.-K. Direct Fidelity Estimation from Few Pauli Measurements. Phys. Rev. Lett. 2011, 106, 230501, DOI: 10.1103/PhysRevLett.106.230501
Google Scholar
There is no corresponding record for this reference.
37
Neill, C. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 2018, 360, 195– 199, DOI: 10.1126/science.aao4309
Google Scholar
There is no corresponding record for this reference.
38
Boixo, S.; Isakov, S. V.; Smelyanskiy, V. N.; Babbush, R.; Ding, N.; Jiang, Z.; Bremner, M. J.; Martinis, J. M.; Neven, H. Characterizing quantum supremacy in near-term devices. Nature Physics 2018, 14, 595– 600
Google Scholar
There is no corresponding record for this reference.
39
Arute, F. Quantum supremacy using a programmable superconducting processor. Nature 2019, 574, 505– 510
Google Scholar
There is no corresponding record for this reference.
40
Kueng, R.; Long, D. M.; Doherty, A. C.; Flammia, S. T. Comparing Experiments to the Fault-Tolerance Threshold. Phys. Rev. Lett. 2016, 117, 170502, DOI: 10.1103/PhysRevLett.117.170502
Google Scholar
There is no corresponding record for this reference.
41
Barnes, J. P.; Trout, C. J.; Lucarelli, D.; Clader, B. D. Quantum error-correction failure distributions: Comparison of coherent and stochastic error models. Phys. Rev. A 2017, 95, 062338 DOI: 10.1103/PhysRevA.95.062338
Google Scholar
There is no corresponding record for this reference.
42
Ng, H. K.; Preskill, J. Fault-tolerant quantum computation versus Gaussian noise. Phys. Rev. A 2009, 79, 032318. DOI: 10.1103/PhysRevA.79.032318
Google Scholar
There is no corresponding record for this reference.
43
Wallman, J.; Granade, C.; Harper, R.; Flammia, S. T. Estimating the coherence of noise. New J. Phys. 2015, 17, 113020, DOI: 10.1088/1367-2630/17/11/113020
Google Scholar
There is no corresponding record for this reference.
44
Sanders, Y. R.; Wallman, J. J.; Sanders, B. C. Bounding quantum gate error rate based on reported average fidelity. New J. Phys. 2016, 18, 012002 DOI: 10.1088/1367-2630/18/1/012002
Google Scholar
There is no corresponding record for this reference.
45
Hashim, A.; Seritan, S.; Proctor, T.; Rudinger, K.; Goss, N.; Naik, R. K.; Kreikebaum, J. M.; Santiago, D. I.; Siddiqi, I. Benchmarking quantum logic operations relative to thresholds for fault tolerance. npj Quantum Information 2023, 9, 109, DOI: 10.1038/s41534-023-00764-y
Google Scholar
There is no corresponding record for this reference.
46
Wallman, J. J.; Emerson, J. Noise tailoring for scalable quantum computation via randomized compiling. Phys. Rev. A 2016, 94, 052325– 052334, DOI: 10.1103/PhysRevA.94.052325
Google Scholar
There is no corresponding record for this reference.
47
Hashim, A.; Naik, R. K.; Morvan, A.; Ville, J.-L.; Mitchell, B.; Kreikebaum, J. M.; Davis, M.; Smith, E.; Iancu, C.; O’Brien, K. P.; Hincks, I.; Wallman, J. J.; Emerson, J.; Siddiqi, I. Randomized Compiling for Scalable Quantum Computing on a Noisy Superconducting Quantum Processor. Phys. Rev. X 2021, 11, 041039 DOI: 10.1103/PhysRevX.11.041039
Google Scholar
There is no corresponding record for this reference.
48
van den Berg, E.; Minev, Z. K.; Kandala, A.; Temme, K. Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors. Nat. Phys. 2023, 19, 1116– 1121, DOI: 10.1038/s41567-023-02042-2
Google Scholar
There is no corresponding record for this reference.
49
Seif, A.; Liao, H.; Tripathi, V.; Krsulich, K.; Malekakhlagh, M.; Amico, M.; Jurcevic, P.; Javadi-Abhari, A. Suppressing Correlated Noise in Quantum Computers via Context-Aware Compiling. 51st Annual International Symposium on Computer Architecture (ISCA) 2024, 310– 324
Google Scholar
There is no corresponding record for this reference.
50
Gross, J. A.; Genois, E.; Debroy, D. M.; Zhang, Y.; Mruczkiewicz, W.; Cian, Z.-P.; Jiang, Z. Characterizing Coherent Errors using Matrix-Element Amplification. npj Quantum Information 2024, 10, 123, DOI: 10.1038/s41534-024-00917-7
Google Scholar
There is no corresponding record for this reference.
51
Bravyi, S.; Englbrecht, M.; König, R.; Peard, N. Correcting coherent errors with surface codes. npj Quantum Information 2018, 4, 55, DOI: 10.1038/s41534-018-0106-y
Google Scholar
There is no corresponding record for this reference.
52
Barenco, A.; Bennett, C. H.; Cleve, R.; DiVincenzo, D. P.; Margolus, N.; Shor, P.; Sleator, T.; Smolin, J. A.; Weinfurter, H. Elementary gates for quantum computation. Phys. Rev. A 1995, 52, 3457– 3467, DOI: 10.1103/PhysRevA.52.3457
Google Scholar
There is no corresponding record for this reference.
53
Erhard, A.; Wallman, J. J.; Postler, L.; Meth, M.; Stricker, R.; Martinez, E. A.; Schindler, P.; Monz, T.; Emerson, J.; Blatt, R. Characterizing large-scale quantum computers via cycle benchmarking. Nat. Commun. 2019, 10, 5347, DOI: 10.1038/s41467-019-13068-7
Google Scholar
There is no corresponding record for this reference.
54
McKay, D. C.; Wood, C. J.; Sheldon, S.; Chow, J. M.; Gambetta, J. M. Efficient Z gates for quantum computing. Phys. Rev. A 2017, 96, 022330 DOI: 10.1103/PhysRevA.96.022330
Google Scholar
There is no corresponding record for this reference.
55
Cross, A. W.; Bishop, L. S.; Sheldon, S.; Nation, P. D.; Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 2019, 100, 032328 DOI: 10.1103/PhysRevA.100.032328
Google Scholar
There is no corresponding record for this reference.
56
Dankert, C.; Cleve, R.; Emerson, J.; Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 2009, 80, 012304 DOI: 10.1103/PhysRevA.80.012304
Google Scholar
There is no corresponding record for this reference.
57
Proctor, T.; Rudinger, K.; Young, K.; Nielsen, E.; Blume-Kohout, R. Measuring the capabilities of quantum computers. Nat. Phys. 2022, 18, 75– 79, DOI: 10.1038/s41567-021-01409-7
Google Scholar
There is no corresponding record for this reference.
58
Cincio, L.; Subası, Y.; Sornborger, A. T; Coles, P. J Learning the quantum algorithm for state overlap. New J. Phys. 2018, 20, 113022, DOI: 10.1088/1367-2630/aae94a
Google Scholar
There is no corresponding record for this reference.
59
Reich, D. M.; Gualdi, G.; Koch, C. P. Optimal Strategies for Estimating the Average Fidelity of Quantum Gates. Phys. Rev. Lett. 2013, 111, 200401– 200406, DOI: 10.1103/PhysRevLett.111.200401
Google Scholar
There is no corresponding record for this reference.
60
Torlai, G.; Mazzola, G.; Carrasquilla, J.; Troyer, M.; Melko, R.; Carleo, G. Neural-network quantum state tomography. Nat. Phys. 2018, 14, 447– 450, DOI: 10.1038/s41567-018-0048-5
Google Scholar
There is no corresponding record for this reference.
61
Koutny, D.; Motka, L.; Hradil, Z.; Rehacek, J.; Sanchez-Soto, L. L. Neural-network quantum state tomography. Phys. Rev. A 2022, 106. DOI: 10.1103/PhysRevA.106.012409
Google Scholar
There is no corresponding record for this reference.
62
Gu, Y.; Zhuang, W.-F.; Chai, X.; Liu, D. E. Benchmarking universal quantum gates via channel spectrum. Nat. Commun. 2023, 14, 5880, DOI: 10.1038/s41467-023-41598-8
Google Scholar
There is no corresponding record for this reference.
63
McKay, D. C.; Hincks, I.; Pritchett, E. J.; Carroll, M.; Govia, L. C. G.; Merkel, S. T. Benchmarking Quantum Processor Performance at Scale. 2023; https://arxiv.org/abs/2311.05933 (accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
64
Breuer, H.-P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press: Oxford, 2002.
Google Scholar
There is no corresponding record for this reference.
65
Kraus, K. States, Effects and Operations; Fundamental Notions of Quantum Theory; Academic, 1983.
Google Scholar
There is no corresponding record for this reference.
66
Nielsen, M. A.; Chuang, I. L. Quantum computation and quantum information; Cambridge University Press, 2010.
Google Scholar
There is no corresponding record for this reference.
67
Jamiołkowski, A. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics 1972, 3, 275– 278, DOI: 10.1016/0034-4877(72)90011-0
Google Scholar
There is no corresponding record for this reference.
68
Choi, M.-D. Completely positive linear maps on complex matrices. Linear Algebra and its Applications 1975, 10, 285– 290, DOI: 10.1016/0024-3795(75)90075-0
Google Scholar
There is no corresponding record for this reference.
69
Uhlmann, A. The “transition probability”in the state space of a ∗-algebra. Reports on Mathematical Physics 1976, 9, 273– 279, DOI: 10.1016/0034-4877(76)90060-4
Google Scholar
There is no corresponding record for this reference.
70
Knill, E.; Laflamme, R. Theory of quantum error-correcting codes. Phys. Rev. A 1997, 55, 900– 911, DOI: 10.1103/PhysRevA.55.900
Google Scholar
There is no corresponding record for this reference.
71
Tripathi, V.; Chen, H.; Levenson-Falk, E.; Lidar, D. A. Modeling Low- and High-Frequency Noise in Transmon Qubits with Resource-Efficient Measurement. PRX Quantum 2024, 5, 010320 DOI: 10.1103/PRXQuantum.5.010320
Google Scholar
There is no corresponding record for this reference.
72
Byrd, M. S.; Lidar, D. A.; Wu, L.-A.; Zanardi, P. Universal leakage elimination. Phys. Rev. A 2005, 71, 052301 DOI: 10.1103/PhysRevA.71.052301
Google Scholar
There is no corresponding record for this reference.
73
Koch, J.; Yu, T. M.; Gambetta, J.; Houck, A. A.; Schuster, D. I.; Majer, J.; Blais, A.; Devoret, M. H.; Girvin, S. M.; Schoelkopf, R. J. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 2007, 76, 042319 DOI: 10.1103/PhysRevA.76.042319
Google Scholar
There is no corresponding record for this reference.
74
Wu, L.-A.; Walther, P.; Lidar, D. A. No-go theorem for passive single-rail linear optical quantum computing. Sci. Rep. 2013, 3, 1394, DOI: 10.1038/srep01394
Google Scholar
There is no corresponding record for this reference.
75
Levine, H. Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons. Physical Review X 2024, 14, 011051– 011072, DOI: 10.1103/PhysRevX.14.011051
Google Scholar
There is no corresponding record for this reference.
76
Grünzweig, T.; Hilliard, A.; McGovern, M.; Andersen, M. F. Near-deterministic preparation of a single atom in an optical microtrap. Nat. Phys. 2010, 6, 951– 954, DOI: 10.1038/nphys1778
Google Scholar
There is no corresponding record for this reference.
77
Wu, Y.; Kolkowitz, S.; Puri, S.; Thompson, J. D. Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays. Nat. Commun. 2022, 13, 4657, DOI: 10.1038/s41467-022-32094-6
Google Scholar
There is no corresponding record for this reference.
78
Varbanov, B. M.; Battistel, F.; Tarasinski, B. M.; Ostroukh, V. P.; O’Brien, T. E.; DiCarlo, L.; Terhal, B. M. Leakage detection for a transmon-based surface code. npj Quantum Information 2020, 6, 102, DOI: 10.1038/s41534-020-00330-w
Google Scholar
There is no corresponding record for this reference.
79
McEwen, M. Removing leakage-induced correlated errors in superconducting quantum error correction. Nat. Commun. 2021, 12, 1761, DOI: 10.1038/s41467-021-21982-y
Google Scholar
There is no corresponding record for this reference.
80
Gourlay, I.; Snowdon, J. F. Concatenated Coding in the Presence of Dephasing. Phys. Rev. A 2000, 62, 022308 DOI: 10.1103/PhysRevA.62.022308
Google Scholar
There is no corresponding record for this reference.
81
Evans, Z. W. E.; Stephens, A. M.; Cole, J. H.; Hollenberg, L. C. L. Error Correction Optimisation in the Presence of X/Z Asymmetry. 2007.
Google Scholar
There is no corresponding record for this reference.
82
Stephens, A. M.; Evans, Z. W. E.; Devitt, S. J.; Hollenberg, L. C. L. Asymmetric Quantum Error Correction via Code Conversion. Phys. Rev. A 2008, 77, 062335 DOI: 10.1103/PhysRevA.77.062335
Google Scholar
There is no corresponding record for this reference.
83
Aliferis, P.; Preskill, J. Fault-tolerant quantum computation against biased noise. Phys. Rev. A 2008, 78, 052331 DOI: 10.1103/PhysRevA.78.052331
Google Scholar
There is no corresponding record for this reference.
84
Stephens, A. M.; Munro, W. J.; Nemoto, K. High-Threshold Topological Quantum Error Correction against Biased Noise. Phys. Rev. A 2013, 88, 060301 DOI: 10.1103/PhysRevA.88.060301
Google Scholar
There is no corresponding record for this reference.
85
Brooks, P.; Preskill, J. Fault-Tolerant Quantum Computation with Asymmetric Bacon-Shor Codes. Phys. Rev. A 2013, 87, 032310 DOI: 10.1103/PhysRevA.87.032310
Google Scholar
There is no corresponding record for this reference.
86
Bonilla Ataides, J. P.; Tuckett, D. K.; Bartlett, S. D.; Flammia, S. T.; Brown, B. J. The XZZX surface code. Nat. Commun. 2021, 12, 2172, DOI: 10.1038/s41467-021-22274-1
Google Scholar
There is no corresponding record for this reference.
87
Chruscinski, D.; Kossakowski, A. Non-Markovian Quantum Dynamics: Local versus Nonlocal. Phys. Rev. Lett. 2010, 104, 070406 DOI: 10.1103/PhysRevLett.104.070406
Google Scholar
There is no corresponding record for this reference.
88
Rivas, Á.; Huelga, S. F.; Plenio, M. B. Quantum non-Markovianity: characterization, quantification and detection. Rep. Prog. Phys. 2014, 77, 094001 DOI: 10.1088/0034-4885/77/9/094001
Google Scholar
There is no corresponding record for this reference.
89
Shabani, A.; Lidar, D. A. Completely positive post-Markovian master equation via a measurement approach. Phys. Rev. A 2005, 71, 020101 DOI: 10.1103/PhysRevA.71.020101
Google Scholar
There is no corresponding record for this reference.
90
Zhang, H.; Pokharel, B.; Levenson-Falk, E.; Lidar, D. Predicting Non-Markovian Superconducting-Qubit Dynamics from Tomographic Reconstruction. Physical Review Applied 2022, 17, 054018 DOI: 10.1103/PhysRevApplied.17.054018
Google Scholar
There is no corresponding record for this reference.
91
Zhang, J.; Vala, J.; Sastry, S.; Whaley, K. B. Geometric theory of nonlocal two-qubit operations. Phys. Rev. A 2003, 67, 042313 DOI: 10.1103/PhysRevA.67.042313
Google Scholar
There is no corresponding record for this reference.
92
Sete, E. A.; Didier, N.; Chen, A. Q.; Kulshreshtha, S.; Manenti, R.; Poletto, S. Parametric-Resonance Entangling Gates with a Tunable Coupler. Phys. Rev. Appl. 2021, 16, 024050 DOI: 10.1103/PhysRevApplied.16.024050
Google Scholar
There is no corresponding record for this reference.
93
Sete, E. A.; Tripathi, V.; Valery, J. A.; Lidar, D.; Mutus, J. Y. Error Budget of a Parametric Resonance Entangling Gate with a Tunable Coupler. Physical Review Applied 2024, 22, 014059 DOI: 10.1103/PhysRevApplied.22.014059
Google Scholar
There is no corresponding record for this reference.
94
Tripathi, V.; Khezri, M.; Korotkov, A. N. Operation and intrinsic error budget of a two-qubit cross-resonance gate. Phys. Rev. A 2019, 100, 012301 DOI: 10.1103/PhysRevA.100.012301
Google Scholar
There is no corresponding record for this reference.
95
Gaikwad, C.; Kowsari, D.; Brame, C.; Song, X.; Zhang, H.; Esposito, M.; Ranadive, A.; Cappelli, G.; Roch, N.; Levenson-Falk, E. M.; Murch, K. W. Entanglement Assisted Probe of the Non-Markovian to Markovian Transition in Open Quantum System Dynamics. Phys. Rev. Lett. 2024, 132, 200401, DOI: 10.1103/PhysRevLett.132.200401
Google Scholar
There is no corresponding record for this reference.
96
Dubi, Y.; Di Ventra, M. Relaxation times in an open interacting two-qubit system. Phys. Rev. A 2009, 79, 012328 DOI: 10.1103/PhysRevA.79.012328
Google Scholar
There is no corresponding record for this reference.
97
Greenbaum, D. Introduction to Quantum Gate Set Tomography. 2015, arXiv:1509.02921 [quant-ph] (accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
98
Brown, W. G.; Eastin, B. Randomized Benchmarking with Restricted Gate Sets. Phys. Rev. A 2018, 97, 062323 DOI: 10.1103/PhysRevA.97.062323
Google Scholar
There is no corresponding record for this reference.
99
Seif, A.; Chen, S.; Majumder, S.; Liao, H.; Wang, D. S.; Malekakhlagh, M.; Javadi-Abhari, A.; Jiang, L.; Minev, Z. K. Entanglement-enhanced learning of quantum processes at scale. 2024; https://arxiv.org/abs/2408.03376 (accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
100
Khan, M. Q.; Dong, W.; Norris, L. M.; Viola, L. Multiaxis quantum noise spectroscopy robust to errors in state preparation and measurement. Physical Review Applied 2024, 22, 024074 DOI: 10.1103/PhysRevApplied.22.024074
Google Scholar
There is no corresponding record for this reference.
101
Merkel, S. T.; Gambetta, J. M.; Smolin, J. A.; Poletto, S.; Córcoles, A. D.; Johnson, B. R.; Ryan, C. A.; Steffen, M. Self-consistent quantum process tomography. Phys. Rev. A 2013, 87, 062119 DOI: 10.1103/PhysRevA.87.062119
Google Scholar
There is no corresponding record for this reference.
102
Magesan, E.; Gambetta, J. M.; Emerson, J. Characterizing quantum gates via randomized benchmarking. Phys. Rev. A 2012, 85, 042311 DOI: 10.1103/PhysRevA.85.042311
Google Scholar
There is no corresponding record for this reference.
103
Aaronson, S.; Gottesman, D. Improved simulation of stabilizer circuits. Phys. Rev. A 2004, 70, 052328 DOI: 10.1103/PhysRevA.70.052328
Google Scholar
There is no corresponding record for this reference.
104
Nielsen, M. A. A simple formula for the average gate fidelity of a quantum dynamical operation. Phys. Lett. A 2002, 303, 249– 252, DOI: 10.1016/S0375-9601(02)01272-0
Google Scholar
There is no corresponding record for this reference.
105
Viola, L.; Knill, E. Random Decoupling Schemes for Quantum Dynamical Control and Error Suppression. Phys. Rev. Lett. 2005, 94, 060502 DOI: 10.1103/PhysRevLett.94.060502
Google Scholar
There is no corresponding record for this reference.
106
Santos, L. F.; Viola, L. Dynamical control of qubit coherence: Random versus deterministic schemes. Phys. Rev. A 2005, 72, 062303 DOI: 10.1103/PhysRevA.72.062303
Google Scholar
There is no corresponding record for this reference.
107
Santos, L. F.; Viola, L. Enhanced Convergence and Robust Performance of Randomized Dynamical Decoupling. Phys. Rev. Lett. 2006, 97, 150501, DOI: 10.1103/PhysRevLett.97.150501
Google Scholar
There is no corresponding record for this reference.
108
Magesan, E.; Gambetta, J. M.; Johnson, B. R.; Ryan, C. A.; Chow, J. M.; Merkel, S. T.; da Silva, M. P.; Keefe, G. A.; Rothwell, M. B.; Ohki, T. A.; Ketchen, M. B.; Steffen, M. Efficient Measurement of Quantum Gate Error by Interleaved Randomized Benchmarking. Phys. Rev. Lett. 2012, 109, 080505 DOI: 10.1103/PhysRevLett.109.080505
Google Scholar
There is no corresponding record for this reference.
109
Gambetta, J. M.; Córcoles, A. D.; Merkel, S. T.; Johnson, B. R.; Smolin, J. A.; Chow, J. M.; Ryan, C. A.; Rigetti, C.; Poletto, S.; Ohki, T. A.; Ketchen, M. B.; Steffen, M. Characterization of Addressability by Simultaneous Randomized Benchmarking. Phys. Rev. Lett. 2012, 109, 240504, DOI: 10.1103/PhysRevLett.109.240504
Google Scholar
There is no corresponding record for this reference.
110
Wallman, J. J.; Barnhill, M.; Emerson, J. Robust characterization of leakage errors. New J. Phys. 2016, 18, 043021 DOI: 10.1088/1367-2630/18/4/043021
Google Scholar
There is no corresponding record for this reference.
111
Wood, C. J.; Gambetta, J. M. Quantification and characterization of leakage errors. Phys. Rev. A 2018, 97, 032306 DOI: 10.1103/PhysRevA.97.032306
Google Scholar
There is no corresponding record for this reference.
112
Fogarty, M. A.; Veldhorst, M.; Harper, R.; Yang, C. H.; Bartlett, S. D.; Flammia, S. T.; Dzurak, A. S. Nonexponential fidelity decay in randomized benchmarking with low-frequency noise. Phys. Rev. A 2015, 92, 022326 DOI: 10.1103/PhysRevA.92.022326
Google Scholar
There is no corresponding record for this reference.
113
Wallman, J. J. Randomized benchmarking with gate-dependent noise. Quantum 2018, 2, 47, DOI: 10.22331/q-2018-01-29-47
Google Scholar
There is no corresponding record for this reference.
114
Proctor, T.; Seritan, S.; Rudinger, K.; Nielsen, E.; Blume-Kohout, R.; Young, K. Scalable Randomized Benchmarking of Quantum Computers Using Mirror Circuits. Phys. Rev. Lett. 2022, 129, 150502, DOI: 10.1103/PhysRevLett.129.150502
Google Scholar
There is no corresponding record for this reference.
115
Helsen, J.; Xue, X.; Vandersypen, L. M. K.; Wehner, S. A new class of efficient randomized benchmarking protocols. npj Quantum Information 2019, 5, 71, DOI: 10.1038/s41534-019-0182-7
Google Scholar
There is no corresponding record for this reference.
116
Helsen, J.; Roth, I.; Onorati, E.; Werner, A.; Eisert, J. General Framework for Randomized Benchmarking. PRX Quantum 2022, 3, 020357 DOI: 10.1103/PRXQuantum.3.020357
Google Scholar
There is no corresponding record for this reference.
117
Arrighi, P.; Patricot, C. On quantum operations as quantum states. Annals of Physics 2004, 311, 26– 52, DOI: 10.1016/j.aop.2003.11.005
Google Scholar
There is no corresponding record for this reference.
118
D’Ariano, G. M.; Lo Presti, P. Quantum Tomography for Measuring Experimentally the Matrix Elements of an Arbitrary Quantum Operation. Phys. Rev. Lett. 2001, 86, 4195– 4198, DOI: 10.1103/PhysRevLett.86.4195
Google Scholar
There is no corresponding record for this reference.
119
Leung, D. W. Choi’s proof as a recipe for quantum process tomography. Journal of Mathematical Physics 2003, 44, 528, DOI: 10.1063/1.1518554
Google Scholar
There is no corresponding record for this reference.
120
Altepeter, J. B.; Branning, D.; Jeffrey, E.; Wei, T. C.; Kwiat, P. G.; Thew, R. T.; O’Brien, J. L.; Nielsen, M. A.; White, A. G. Ancilla-Assisted Quantum Process Tomography. Phys. Rev. Lett. 2003, 90, 193601, DOI: 10.1103/PhysRevLett.90.193601
Google Scholar
There is no corresponding record for this reference.
121
D’Ariano, G. M.; Lo Presti, P. Imprinting Complete Information about a Quantum Channel on its Output State. Phys. Rev. Lett. 2003, 91, 047902 DOI: 10.1103/PhysRevLett.91.047902
Google Scholar
There is no corresponding record for this reference.
122
Mohseni, M.; Lidar, D. A. Direct Characterization of Quantum Dynamics. Phys. Rev. Lett. 2006, 97, 170501, DOI: 10.1103/PhysRevLett.97.170501
Google Scholar
There is no corresponding record for this reference.
123
Mohseni, M.; Lidar, D. A. Direct characterization of quantum dynamics: General theory. Phys. Rev. A 2007, 75, 062331 DOI: 10.1103/PhysRevA.75.062331
Google Scholar
There is no corresponding record for this reference.
124
Mohseni, M.; Rezakhani, A. T.; Aspuru-Guzik, A. Direct estimation of single- and two-qubit Hamiltonians and relaxation rates. Phys. Rev. A 2008, 77, 042320. DOI: 10.1103/PhysRevA.77.042320
Google Scholar
There is no corresponding record for this reference.
125
Graham, T. M.; Barreiro, J. T.; Mohseni, M.; Kwiat, P. G. Hyperentanglement-Enabled Direct Characterization of Quantum Dynamics. Phys. Rev. Lett. 2013, 110, 060404 DOI: 10.1103/PhysRevLett.110.060404
Google Scholar
There is no corresponding record for this reference.
126
Nigg, D.; Barreiro, J. T.; Schindler, P.; Mohseni, M.; Monz, T.; Chwalla, M.; Hennrich, M.; Blatt, R. Experimental Characterization of Quantum Dynamics Through Many-Body Interactions. Phys. Rev. Lett. 2013, 110, 060403 DOI: 10.1103/PhysRevLett.110.060403
Google Scholar
There is no corresponding record for this reference.
127
Gottesman, D. Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 1996, 54, 1862, DOI: 10.1103/PhysRevA.54.1862
Google Scholar
There is no corresponding record for this reference.
128
Blume-Kohout, R.; Gamble, J. K.; Nielsen, E.; Rudinger, K.; Mizrahi, J.; Fortier, K.; Maunz, P. Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography. Nat. Commun. 2017, 8, 14485, DOI: 10.1038/ncomms14485
Google Scholar
There is no corresponding record for this reference.
129
Rol, M. A.; Bultink, C. C.; O’Brien, T. E.; de Jong, S. R.; Theis, L. S.; Fu, X.; Luthi, F.; Vermeulen, R. F. L.; de Sterke, J. C.; Bruno, A.; Deurloo, D.; Schouten, R. N.; Wilhelm, F. K.; DiCarlo, L. Restless Tuneup of High-Fidelity Qubit Gates. Phys. Rev. Appl. 2017, 7, 041001 DOI: 10.1103/PhysRevApplied.7.041001
Google Scholar
There is no corresponding record for this reference.
130
Mavadia, S.; Edmunds, C. L.; Hempel, C.; Ball, H.; Roy, F.; Stace, T. M.; Biercuk, M. J. Experimental quantum verification in the presence of temporally correlated noise. npj Quantum Information 2018, 4, 7, DOI: 10.1038/s41534-017-0052-0
Google Scholar
There is no corresponding record for this reference.
131
Hempel, C.; Maier, C.; Romero, J.; McClean, J.; Monz, T.; Shen, H.; Jurcevic, P.; Lanyon, B. P.; Love, P.; Babbush, R.; Aspuru-Guzik, A.; Blatt, R.; Roos, C. F. Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator. Phys. Rev. X 2018, 8, 031022 DOI: 10.1103/PhysRevX.8.031022
Google Scholar
There is no corresponding record for this reference.
132
Song, C.; Cui, J.; Wang, H.; Hao, J.; Feng, H.; Li, Y. Quantum computation with universal error mitigation on a superconducting quantum processor. Science Advances 2019, 5, eaaw5686 DOI: 10.1126/sciadv.aaw5686
Google Scholar
There is no corresponding record for this reference.
133
Cincio, L.; Rudinger, K.; Sarovar, M.; Coles, P. J. Machine Learning of Noise-Resilient Quantum Circuits. PRX Quantum 2021, 2, 010324 DOI: 10.1103/PRXQuantum.2.010324
Google Scholar
There is no corresponding record for this reference.
134
Hong, S. S.; Papageorge, A. T.; Sivarajah, P.; Crossman, G.; Didier, N.; Polloreno, A. M.; Sete, E. A.; Turkowski, S. W.; da Silva, M. P.; Johnson, B. R. Demonstration of a parametrically activated entangling gate protected from flux noise. Phys. Rev. A 2020, 101, 012302 DOI: 10.1103/PhysRevA.101.012302
Google Scholar
There is no corresponding record for this reference.
135
Joshi, M. K.; Elben, A.; Vermersch, B.; Brydges, T.; Maier, C.; Zoller, P.; Blatt, R.; Roos, C. F. Quantum Information Scrambling in a Trapped-Ion Quantum Simulator with Tunable Range Interactions. Phys. Rev. Lett. 2020, 124, 240505, DOI: 10.1103/PhysRevLett.124.240505
Google Scholar
There is no corresponding record for this reference.
136
Zhang, S.; Lu, Y.; Zhang, K.; Chen, W.; Li, Y.; Zhang, J.-N.; Kim, K. Error-mitigated quantum gates exceeding physical fidelities in a trapped-ion system. Nat. Commun. 2020, 11, 587, DOI: 10.1038/s41467-020-14376-z
Google Scholar
There is no corresponding record for this reference.
137
Ware, M.; Ribeill, G.; Ristè, D.; Ryan, C. A.; Johnson, B.; da Silva, M. P. Experimental Pauli-frame randomization on a superconducting qubit. Phys. Rev. A 2021, 103, 042604 DOI: 10.1103/PhysRevA.103.042604
Google Scholar
There is no corresponding record for this reference.
138
Dehollain, J. P.; Muhonen, J. T.; Blume-Kohout, R.; Rudinger, K. M.; Gamble, J. K.; Nielsen, E.; Laucht, A.; Simmons, S.; Kalra, R.; Dzurak, A. S.; Morello, A. Optimization of a solid-state electron spin qubit using gate set tomography. New J. Phys. 2016, 18, 103018, DOI: 10.1088/1367-2630/18/10/103018
Google Scholar
There is no corresponding record for this reference.
139
Kitaev, A.; Shen, A.; Vyalyi, M. Classical and Quantum Computation; Graduate studies in mathematics; American Mathematical Society, 2002.
Google Scholar
There is no corresponding record for this reference.
140
Scott, A. J. Tight informationally complete quantum measurements. Journal of Physics A: Mathematical and General 2006, 39, 13507, DOI: 10.1088/0305-4470/39/43/009
Google Scholar
There is no corresponding record for this reference.
141
Nielsen, E.; Gamble, J. K.; Rudinger, K.; Scholten, T.; Young, K.; Blume-Kohout, R. Gate Set Tomography. Quantum 2021, 5, 557, DOI: 10.22331/q-2021-10-05-557
Google Scholar
There is no corresponding record for this reference.
142
Householder, A. The Theory of Matrices in Numerical Analysis; Dover Books on Mathematics; Dover Publications, 2013.
Google Scholar
There is no corresponding record for this reference.
143
Nielsen, E.; Rudinger, K.; Proctor, T.; Russo, A.; Young, K.; Blume-Kohout, R. Probing quantum processor performance with pyGSTi. Quantum Science and Technology 2020, 5, 044002 DOI: 10.1088/2058-9565/ab8aa4
Google Scholar
There is no corresponding record for this reference.
144
Proctor, T.; Revelle, M.; Nielsen, E.; Rudinger, K.; Lobser, D.; Maunz, P.; Blume-Kohout, R.; Young, K. Detecting and tracking drift in quantum information processors. Nat. Commun. 2020, 11, 5396, DOI: 10.1038/s41467-020-19074-4
Google Scholar
There is no corresponding record for this reference.
145
Rudinger, K. M.; Ostrove, C. I.; Seritan, S. K.; Grace, M. D.; Nielsen, E.; Blume-Kohout, R. J.; Young, K. C. Two-Qubit Gate Set Tomography with Fewer Circuits. 2023; https://arxiv.org/abs/2307.15767 (accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
146
Horodecki, M.; Horodecki, P.; Horodecki, R. General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A 1999, 60, 1888– 1898, DOI: 10.1103/PhysRevA.60.1888
Google Scholar
There is no corresponding record for this reference.
147
Lu, D.; Li, H.; Trottier, D.-A.; Li, J.; Brodutch, A.; Krismanich, A. P.; Ghavami, A.; Dmitrienko, G. I.; Long, G.; Baugh, J.; Laflamme, R. Experimental Estimation of Average Fidelity of a Clifford Gate on a 7-qubit Quantum Processor. Phys. Rev. Lett. 2015, 114, 140505, DOI: 10.1103/PhysRevLett.114.140505
Google Scholar
There is no corresponding record for this reference.
148
Fedorov, A.; Steffen, L.; Baur, M.; da Silva, M. P.; Wallraff, A. Implementation of a Toffoli Gate with Superconducting Circuits. Nature 2012, 481, 170– 172, DOI: 10.1038/nature10713
Google Scholar
There is no corresponding record for this reference.
149
Wallman, J. J.; Flammia, S. T. Randomized benchmarking with confidence. New J. Phys. 2014, 16, 103032, DOI: 10.1088/1367-2630/16/10/103032
Google Scholar
There is no corresponding record for this reference.
150
Roth, I.; Kueng, R.; Kimmel, S.; Liu, Y.-K.; Gross, D.; Eisert, J.; Kliesch, M. Recovering quantum gates from few average gate fidelities. Phys. Rev. Lett. 2018, 121, 170502, DOI: 10.1103/PhysRevLett.121.170502
Google Scholar
There is no corresponding record for this reference.
151
Kalev, A.; Kyrillidis, A.; Linke, N. M. Validating and certifying stabilizer states. Phys. Rev. A 2019, 99, 042337 DOI: 10.1103/PhysRevA.99.042337
Google Scholar
There is no corresponding record for this reference.
152
Seshadri, A.; Ringbauer, M.; Spainhour, J.; Monz, T.; Becker, S. Theory of versatile fidelity estimation with confidence. Phys. Rev. A 2024, 110, 012431 DOI: 10.1103/PhysRevA.110.012431
Google Scholar
There is no corresponding record for this reference.
153
Seshadri, A.; Ringbauer, M.; Spainhour, J.; Blatt, R.; Monz, T.; Becker, S. Versatile Fidelity Estimation with Confidence. Phys. Rev. Lett. 2024, 133, 020402 DOI: 10.1103/PhysRevLett.133.020402
Google Scholar
There is no corresponding record for this reference.
154
Kosut, R. L.; Rabitz, H.; Grace, M. D. Adaptive quantum control via direct fidelity estimation and indirect model-based parametric process tomography. 52nd IEEE Conference on Decision and Control. Firenze, 2013; pp 1247– 1252.
Google Scholar
There is no corresponding record for this reference.
155
Lu, Y.; Sim, J. Y.; Suzuki, J.; Englert, B.-G.; Ng, H. K. Direct estimation of minimum gate fidelity. Phys. Rev. A 2020, 102, 022410 DOI: 10.1103/PhysRevA.102.022410
Google Scholar
There is no corresponding record for this reference.
156
Liu, Y.; Otten, M.; Bassirianjahromi, R.; Jiang, L.; Fefferman, B. Benchmarking near-term quantum computers via random circuit sampling. 2022; http://arxiv.org/abs/2105.05232, arXiv: 2105.05232 [quant-ph] (accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
157
Elements of Information Theory; John Wiley & Sons, Ltd, 2005; Chapter 4, pp 71– 101.
Google Scholar
There is no corresponding record for this reference.
158
Ledoux, M. The Concentration of Measure Phenomenon; American Mathematical Soc., 2001; Google-Books-ID: rRKChz4Om6gC.
Google Scholar
There is no corresponding record for this reference.
159
Boixo, S.; Smelyanskiy, V. N.; Neven, H. Fourier analysis of sampling from noisy chaotic quantum circuits. 2017; http://arxiv.org/abs/1708.01875, arXiv: 1708.01875 [quant-ph] (accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
160
Choi, J.; Shaw, A. L.; Madjarov, I. S.; Xie, X.; Finkelstein, R.; Covey, J. P.; Cotler, J. S.; Mark, D. K.; Huang, H.-Y.; Kale, A.; Pichler, H.; Brandão, F. G. S. L.; Choi, S.; Endres, M. Preparing random states and benchmarking with many-body quantum chaos. Nature 2023, 613, 468– 473, DOI: 10.1038/s41586-022-05442-1
Google Scholar
There is no corresponding record for this reference.
161
Rinott, Y.; Shoham, T.; Kalai, G. Statistical Aspects of the Quantum Supremacy Demonstration. Statistical Science 2022, 37, 322– 347
Google Scholar
There is no corresponding record for this reference.
162
Dalzell, A. M.; Hunter-Jones, N.; Brandão, F. G. S. L. Random Quantum Circuits Anticoncentrate in Log Depth. PRX Quantum 2022, 3, 010333
Google Scholar
There is no corresponding record for this reference.
163
Huang, C.; Zhang, F.; Newman, M.; Cai, J.; Gao, X.; Tian, Z.; Wu, J.; Xu, H.; Yu, H.; Yuan, B.; Szegedy, M.; Shi, Y.; Chen, J. Classical Simulation of Quantum Supremacy Circuits. arXiv 2020 (accessed 2025-02-23). DOI: 10.48550/arXiv.2005.06787
Google Scholar
There is no corresponding record for this reference.
164
Zlokapa, A.; Villalonga, B.; Boixo, S.; Lidar, D. A. Boundaries of quantum supremacy via random circuit sampling. npj Quantum Information 2023, 9, 36, DOI: 10.1038/s41534-023-00703-x
Google Scholar
There is no corresponding record for this reference.
165
Kechedzhi, K.; Isakov, S. V.; Mandrà, S.; Villalonga, B.; Mi, X.; Boixo, S.; Smelyanskiy, V. Effective quantum volume, fidelity and computational cost of noisy quantum processing experiments. Future Generation Computer Systems 2024, 153, 431– 441, DOI: 10.1016/j.future.2023.12.002
Google Scholar
There is no corresponding record for this reference.
166
Arute, F. Quantum supremacy using a programmable superconducting processor. Nature 2019, 574, 505– 510, DOI: 10.1038/s41586-019-1666-5
Google Scholar
There is no corresponding record for this reference.
167
Harrow, A. W.; Mehraban, S. Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates. Communications in Mathematical Physics 2023, 401, 1531– 1626, DOI: 10.1007/s00220-023-04675-z
Google Scholar
There is no corresponding record for this reference.
168
Haferkamp, J.; Montealegre-Mora, F.; Heinrich, M.; Eisert, J.; Gross, D.; Roth, I. Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates. Communications in Mathematical Physics 2023, 397, 995– 1041, DOI: 10.1007/s00220-022-04507-6
Google Scholar
There is no corresponding record for this reference.
169
Gao, X.; Kalinowski, M.; Chou, C.-N.; Lukin, M. D.; Barak, B.; Choi, S. Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage. PRX Quantum 2024, 5, 010334 DOI: 10.1103/PRXQuantum.5.010334
Google Scholar
There is no corresponding record for this reference.
170
Tripathi, V.; Kowsari, D.; Saurav, K.; Zhang, H.; M.Levenson-Falk, E.; Lidar, D. A. Benchmarking quantum gates and circuits. 2024; https://arxiv.org/abs/2407.09942 (accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
171
Burnett, J. J.; Bengtsson, A.; Scigliuzzo, M.; Niepce, D.; Kudra, M.; Delsing, P.; Bylander, J. Decoherence benchmarking of superconducting qubits. npj Quantum Information 2019, 5, 54, DOI: 10.1038/s41534-019-0168-5
Google Scholar
There is no corresponding record for this reference.
172
Hahn, E. L. Spin Echoes. Phys. Rev. 1950, 80, 580– 594, DOI: 10.1103/PhysRev.80.580
Google Scholar
There is no corresponding record for this reference.
173
Carr, H. Y.; Purcell, E. M. Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments. Phys. Rev. 1954, 94, 630– 638, DOI: 10.1103/PhysRev.94.630
Google Scholar
There is no corresponding record for this reference.
174
Meiboom, S.; Gill, D. Modified Spin?Echo Method for Measuring Nuclear Relaxation Times. Rev. Sci. Instrum. 1958, 29, 688– 691, DOI: 10.1063/1.1716296
Google Scholar
There is no corresponding record for this reference.
175
Viola, L.; Lloyd, S. Dynamical suppression of decoherence in two-state quantum systems. Phys. Rev. A 1998, 58, 2733– 2744, DOI: 10.1103/PhysRevA.58.2733
Google Scholar
There is no corresponding record for this reference.
176
Uhrig, G. S. Keeping a Quantum Bit Alive by Optimized π-Pulse Sequences. Phys. Rev. Lett. 2007, 98, 100504, DOI: 10.1103/PhysRevLett.98.100504
Google Scholar
There is no corresponding record for this reference.
177
Suter, D.; Álvarez, G. A. Colloquium: Protecting quantum information against environmental noise. Rev. Mod. Phys. 2016, 88, 041001 DOI: 10.1103/RevModPhys.88.041001
Google Scholar
There is no corresponding record for this reference.
178
Jurcevic, P. Demonstration of Quantum Volume 64 on a Superconducting Quantum Computing System. Quantum Sci. Technol. 2021, 6, 025020 DOI: 10.1088/2058-9565/abe519
Google Scholar
There is no corresponding record for this reference.
179
Pokharel, B.; Lidar, D. A. Demonstration of Algorithmic Quantum Speedup. Phys. Rev. Lett. 2023, 130, 210602, DOI: 10.1103/PhysRevLett.130.210602
Google Scholar
There is no corresponding record for this reference.
180
Tripathi, V.; Chen, H.; Khezri, M.; Yip, K.-W.; Levenson-Falk, E.; Lidar, D. A. Suppression of Crosstalk in Superconducting Qubits Using Dynamical Decoupling. Phys. Rev. Appl. 2022, 18, 024068 DOI: 10.1103/PhysRevApplied.18.024068
Google Scholar
There is no corresponding record for this reference.
181
Zhou, Z.; Sitler, R.; Oda, Y.; Schultz, K.; Quiroz, G. Quantum Crosstalk Robust Quantum Control. Phys. Rev. Lett. 2023, 131, 210802, DOI: 10.1103/PhysRevLett.131.210802
Google Scholar
There is no corresponding record for this reference.
182
Bäumer, E.; Tripathi, V.; Wang, D. S.; Rall, P.; Chen, E. H.; Majumder, S.; Seif, A.; Minev, Z. K. Efficient Long-Range Entanglement using Dynamic Circuits. 2023.
Google Scholar
There is no corresponding record for this reference.
183
Shirizly, L.; Misguich, G.; Landa, H. Dissipative Dynamics of Graph-State Stabilizers with Superconducting Qubits. Phys. Rev. Lett. 2024, 132, 010601 DOI: 10.1103/PhysRevLett.132.010601
Google Scholar
There is no corresponding record for this reference.
184
Niu, S.; Todri-Sanial, A.; Bronn, N. T. Multi-qubit dynamical decoupling for enhanced crosstalk suppression. Quantum Science and Technology 2024, 9, 045003 DOI: 10.1088/2058-9565/ad5a37
Google Scholar
There is no corresponding record for this reference.
185
Evert, B.; Izquierdo, Z. G.; Sud, J.; Hu, H.-Y.; Grabbe, S.; Rieffel, E. G.; Reagor, M. J.; Wang, Z. Syncopated Dynamical Decoupling for Suppressing Crosstalk in Quantum Circuits. 2024; https://arxiv.org/abs/2403.07836 (accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
186
Brown, A. F.; Lidar, D. A. Efficient Chromatic-Number-Based Multi-Qubit Decoherence and Crosstalk Suppression. 2024. https://arxiv.org/abs/2406.13901(accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
187
Martinis, J. M.; Geller, M. R. Fast adiabatic qubit gates using only σ_z control. Phys. Rev. A 2014, 90, 022307 DOI: 10.1103/PhysRevA.90.022307
Google Scholar
There is no corresponding record for this reference.
188
Krantz, P.; Kjaergaard, M.; Yan, F.; Orlando, T. P.; Gustavsson, S.; Oliver, W. D. A quantum engineer’s guide to superconducting qubits. Applied Physics Reviews 2019, 6, 021318 DOI: 10.1063/1.5089550
Google Scholar
There is no corresponding record for this reference.
189
Genov, G. T.; Schraft, D.; Vitanov, N. V.; Halfmann, T. Arbitrarily Accurate Pulse Sequences for Robust Dynamical Decoupling. Phys. Rev. Lett. 2017, 118, 133202, DOI: 10.1103/PhysRevLett.118.133202
Google Scholar
There is no corresponding record for this reference.
190
Motzoi, F.; Gambetta, J. M.; Rebentrost, P.; Wilhelm, F. K. Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits. Phys. Rev. Lett. 2009, 103, 110501, DOI: 10.1103/PhysRevLett.103.110501
Google Scholar
There is no corresponding record for this reference.
191
Chow, J. M.; Gambetta, J. M.; Tornberg, L.; Koch, J.; Bishop, L. S.; Houck, A. A.; Johnson, B. R.; Frunzio, L.; Girvin, S. M.; Schoelkopf, R. J. Randomized Benchmarking and Process Tomography for Gate Errors in a Solid-State Qubit. Phys. Rev. Lett. 2009, 102, 090502 DOI: 10.1103/PhysRevLett.102.090502
Google Scholar
There is no corresponding record for this reference.
192
Albash, T.; Boixo, S.; Lidar, D. A.; Zanardi, P. Quantum adiabatic Markovian master equations. New J. of Phys. 2012, 14, 123016, DOI: 10.1088/1367-2630/14/12/123016
Google Scholar
There is no corresponding record for this reference.
193
Dann, R.; Levy, A.; Kosloff, R. Time-dependent Markovian quantum master equation. Phys. Rev. A 2018, 98, 052129 DOI: 10.1103/PhysRevA.98.052129
Google Scholar
There is no corresponding record for this reference.
194
Campos Venuti, L.; Lidar, D. A. Error reduction in quantum annealing using boundary cancellation: Only the end matters. Phys. Rev. A 2018, 98, 022315 DOI: 10.1103/PhysRevA.98.022315
Google Scholar
There is no corresponding record for this reference.
195
Mozgunov, E.; Lidar, D. Completely positive master equation for arbitrary driving and small level spacing. Quantum 2020, 4, 227, DOI: 10.22331/q-2020-02-06-227
Google Scholar
There is no corresponding record for this reference.
196
Nathan, F.; Rudner, M. S. Universal Lindblad equation for open quantum systems. Phys. Rev. B 2020, 102, 115109, DOI: 10.1103/PhysRevB.102.115109
Google Scholar
There is no corresponding record for this reference.
197
Davidović, D. Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation. Quantum 2020, 4, 326, DOI: 10.22331/q-2020-09-21-326
Google Scholar
There is no corresponding record for this reference.
198
Davidović, D. Geometric-arithmetic master equation in large and fast open quantum systems. Journal of Physics A: Mathematical and Theoretical 2022, 55, 455301, DOI: 10.1088/1751-8121/ac9f30
Google Scholar
There is no corresponding record for this reference.
199
Kasatkin, V.; Gu, L.; Lidar, D. A. Which differential equations correspond to the Lindblad equation?. Physical Review Research 2023, 5, 043163 DOI: 10.1103/PhysRevResearch.5.043163
Google Scholar
There is no corresponding record for this reference.
200
Perko, L. Differential Equations and Dynamical Systems; Springer New York: New York, NY, 2001; pp 1– 63.
Google Scholar
There is no corresponding record for this reference.
201
Geerlings, K.; Leghtas, Z.; Pop, I. M.; Shankar, S.; Frunzio, L.; Schoelkopf, R. J.; Mirrahimi, M.; Devoret, M. H. Demonstrating a Driven Reset Protocol for a Superconducting Qubit. Phys. Rev. Lett. 2013, 110, 120501, DOI: 10.1103/PhysRevLett.110.120501
Google Scholar
There is no corresponding record for this reference.
202
Goss, N.; Morvan, A.; Marinelli, B.; Mitchell, B. K.; Nguyen, L. B.; Naik, R. K.; Chen, L.; Jünger, C.; Kreikebaum, J. M.; Santiago, D. I.; Wallman, J. J.; Siddiqi, I. High-fidelity qutrit entangling gates for superconducting circuits. Nat. Commun. 2022, 13, 7481, DOI: 10.1038/s41467-022-34851-z
Google Scholar
There is no corresponding record for this reference.
203
Tripathi, V.; Goss, N.; Vezvaee, A.; Nguyen, L. B.; Siddiqi, I.; Lidar, D. A. Qudit Dynamical Decoupling on a Superconducting Quantum Processor. Phys. Rev. Lett. 2025, 134, 050601 DOI: 10.1103/PhysRevLett.134.050601
Google Scholar
There is no corresponding record for this reference.
204
Krishnan Vijayan, M.; Paler, A.; Gavriel, J.; Myers, C. R.; Rohde, P. P.; Devitt, S. J. Compilation of algorithm-specific graph states for quantum circuits. Quantum Science and Technology 2024, 9, 025005 DOI: 10.1088/2058-9565/ad1f39
Google Scholar
There is no corresponding record for this reference.
205
Beverland, M. E.; Murali, P.; Troyer, M.; Svore, K. M.; Hoefler, T.; Kliuchnikov, V.; Low, G. H.; Soeken, M.; Sundaram, A.; Vaschillo, A. Assessing requirements to scale to practical quantum advantage. 2022. https://arxiv.org/abs/2211.07629 (accessed 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
206
Agrawal, A. A.; Job, J.; Wilson, T. L.; Saadatmand, S. N.; Hodson, M. J.; Mutus, J. Y.; Caesura, A.; Johnson, P. D.; Elenewski, J. E.; Morrell, K. J.; Kemper, A. F. Quantifying fault tolerant simulation of strongly correlated systems using the Fermi-Hubbard model. 2024; https://arxiv.org/abs/2406.06511(accessed: 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
207
Barends, R. Coherent Josephson Qubit Suitable for Scalable Quantum Integrated Circuits. Phys. Rev. Lett. 2013, 111, 080502 DOI: 10.1103/PhysRevLett.111.080502
Google Scholar
There is no corresponding record for this reference.
208
Shanto, S.; Kuo, A.; Miyamoto, C.; Zhang, H.; Maurya, V.; Vlachos, E.; Hecht, M.; Shum, C. W.; Levenson-Falk, E. SQuADDS: A Validated Design Database and Simulation Workflow for Superconducting Qubit Design. Quantum 2024, 8, 1465, DOI: 10.22331/q-2024-09-09-1465
Google Scholar
There is no corresponding record for this reference.
209
Shah, P. A. . qiskit-community/qiskit-metal: Qiskit Metal 0.1.5a1 (2023). 2023; DOI: 10.5281/zenodo.8061781 (accessed 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
210
Minev, Z. K.; Leghtas, Z.; Reinhold, P.; Mundhada, S. O.; Diringer, A.; Hillel, D. C.; Wang, D. Z.-R.; Facchini, M.; Shah, P. A.; Devoret, M. pyEPR: The energy-participation-ratio (EPR) open- source framework for quantum device design. 2021; DOI: 10.5281/zenodo.4744448 , https://github.com/zlatko-minev/pyEPR; https://pyepr-docs.readthedocs.io/en/latest/ (accessed 2025-02-23).
Google Scholar
There is no corresponding record for this reference.
211
Chen, Z. Measuring and Suppressing Quantum State Leakage in a Superconducting Qubit. Phys. Rev. Lett. 2016, 116, 020501 DOI: 10.1103/PhysRevLett.116.020501
Google Scholar
There is no corresponding record for this reference.

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Abstract
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Figure 1
Figure 1. Three different QPT methods. (a) Schematic of SQPT. An ensemble of states {ρ_j} is prepared and each state is subjected to the map $E$ followed by measurements {E_m}. (b) Schematic of separable AAPT. A “faithful” input state ρ is subjected to the map $E \otimes I$ . The operators {E_j} are measured on the joint system, which results in the required joint probability distributions or expectation values. (c) Schematic of DCQD. The system and the ancilla are prepared in one of the input states as in Table 1 and are subjected to the map $E \otimes I$ . The joint system is measured in the Bell-state basis (Bell State Measurement).
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Figure 2
Figure 2. Gate set tomography circuit consisting of the fiducial gates F_i (F_j) as well as the measurement operators M and the native gate G_k performed on the initial state ρ. Figure adapted from ref (141). Copyright 2021 Quantum.
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Figure 3
Figure 3. Schematic of DFE: To estimate the state fidelity, expectations of random Pauli strings P_j are calculated for the input state ρ. The distribution of the random Pauli strings depends on the state with respect to which the fidelity is being calculated.
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Figure 4
Figure 4. RCS benchmarking circuit: layers of 2-qubit gates (white boxes) are interleaved with Haar random 1-qubit gates (blue boxes), followed by measurement in the computational basis at the end. Adapted from Figure 1 of ref (156). Copyright 2021 arXiv.
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Figure 5
Figure 5. Schematic representations of the gate sequences for (a) randomized benchmarking (RB) and (b) deterministic benchmarking (DB), respectively. RB uses N random Clifford gates C_i, followed by a single recovery gate C_r = C_N^†···C₁^†. DB prepares an initial state via U|0⟩, applies n repetitions of a fixed, deterministic two-pulse sequence P₁P₂, and unprepares the initial state via U^†.
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Figure 6
Figure 6. Empirical (F̃, symbols), analytical fit (F, solid curves), and Lindblad-based (F̅, dashed curves) fidelities for the initial states |1⟩ under free evolution and |+⟩ under various gate sequences P₁P₂. The analytical fit yields T₁ = 23.36 ± 0.40 μs, T₂ = 44.13 ± 2.49 μs, δθ = 0.398 ± 0.004°, and δϕ = 0.426 ± 0.004°. With the exception of the XX (orange) and YY (green) cases, the Lindblad-based results are indistinguishable from the analytical fits.
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Figure 7
Figure 7. Randomized benchmarking measurement for the same gate calibration parameters used in Figure 6. The error bars represent the deviation of the averaged benchmarking results across 30 distinct random sequences with a maximum number of Cliffords (N_Cliffords) of 700, which are smaller than the marker sizes.
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Figure 8
Figure 8. Experimental fidelities F̃^|+⟩ showing the very different sensitivity of DB and RB to coherent errors. (a) δθ = 0°, δϕ = 0°, (b) δθ = 0°, δϕ = 0.893 ± 0.002°, (c) δθ = 0.932 ± 0.007°, δϕ = 0°, (d) δθ = 0.995 ± 0.009°, δϕ = 0.90 ± 0.002°. These angles are obtained from fits using eq 76. Dashed curves are the numerically computed fidelities F^|+⟩ from the open system Lindblad model using the DB parameters above along with the corresponding T₁ and T₂. The UR6 sequence suppresses even large coherent errors.
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Figure 9
Figure 9. Asymmetry in the decay pattern of DB sequences due to the interplay of gates and T₁. Fidelity decay for (a) XX and (b) YY applied to two pairs of orthogonal initial states. Dashed curves represent the Lindblad master equation simulation results with parameters obtained by fitting eq 76.
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Figure 10
Figure 10. Simulated survival fidelity of the |+⟩ state after repetitions of the XX sequence, where each gate is a cosine pulse length of 20 ns (blue circles) or 10 ns (orange squares). A third state in the simulation represents leakage, with a detuning (anharmonicity) of −150 MHz relative to the qubit transition. Both the amplitude and frequency of oscillations increase as the gate duration is decreased and leakage becomes more significant.
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References
This article references 211 other publications.
1. 1
  Bharti, K.; Cervera-Lierta, A.; Kyaw, T. H.; Haug, T.; Alperin-Lea, S.; Anand, A.; Degroote, M.; Heimonen, H.; Kottmann, J. S.; Menke, T.; Mok, W.-K.; Sim, S.; Kwek, L.-C.; Aspuru-Guzik, A. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 2022, 94, 015004 DOI: 10.1103/RevModPhys.94.015004
  There is no corresponding record for this reference.
2. 2
  Dalzell, A. M.; McArdle, S.; Berta, M.; Bienias, P.; Chen, C.-F.; Gilyén, A.; Hann, C. T.; Kastoryano, M. J.; Khabiboulline, E. T.; Kubica, A.; Salton, G.; Wang, S.; Brandão, F. G. S. L. Quantum algorithms: A survey of applications and end-to-end complexities. 2023; https://arxiv.org/abs/2310.03011 (accessed 2025-02-23).
  There is no corresponding record for this reference.
3. 3
  Shor, P. Fault-tolerant quantum computation. Proceedings of 37th Conference on Foundations of Computer Science. 1996; pp 56– 65.
  There is no corresponding record for this reference.
4. 4
  Aharonov, D.; Ben-Or, M. Fault tolerant quantum computation with constant error. Proceedings of 29th Annual ACM Symposium on Theory of Computing (STOC). New York, NY, 1997; p 176.
  There is no corresponding record for this reference.
5. 5
  Knill, E.; Laflamme, R.; Zurek, W. H. Resilient Quantum Computation. Science 1998, 279, 342– 345, DOI: 10.1126/science.279.5349.342
  There is no corresponding record for this reference.
6. 6
  Aliferis, P.; Gottesman, D.; Preskill, J. Quantum accuracy threshold for concatenated distance-3 codes. Quant. Inf. Comput. 2006, 6, 97, DOI: 10.26421/QIC6.2-1
  There is no corresponding record for this reference.
7. 7
  Campbell, E. T.; Terhal, B. M.; Vuillot, C. Roads towards fault-tolerant universal quantum computation. Nature 2017, 549, 172, DOI: 10.1038/nature23460
  There is no corresponding record for this reference.
8. 8
  Reiher, M.; Wiebe, N.; Svore, K. M.; Wecker, D.; Troyer, M. Elucidating reaction mechanisms on quantum computers. Proc. Natl. Acad. Sci. U. S. A. 2017, 114, 7555– 7560, DOI: 10.1073/pnas.1619152114
  There is no corresponding record for this reference.
9. 9
  Kassal, I.; Whitfield, J. D.; Perdomo-Ortiz, A.; Yung, M.-H.; Aspuru-Guzik, A. Simulating Chemistry Using Quantum Computers. Annu. Rev. Phys. Chem. 2011, 62, 185– 207, DOI: 10.1146/annurev-physchem-032210-103512
  There is no corresponding record for this reference.
10. 10
  Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2018, 2, 79, DOI: 10.22331/q-2018-08-06-79
  There is no corresponding record for this reference.
11. 11
  O’Malley, P. J. J. Scalable Quantum Simulation of Molecular Energies. Physical Review X 2016, 6, 031007 DOI: 10.1103/PhysRevX.6.031007
  There is no corresponding record for this reference.
12. 12
  Kandala, A.; Mezzacapo, A.; Temme, K.; Takita, M.; Brink, M.; Chow, J. M.; Gambetta, J. M. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 2017, 549, 242, DOI: 10.1038/nature23879
  There is no corresponding record for this reference.
13. 13
  Lidar, D. A.; Wang, H. Calculating the thermal rate constant with exponential speedup on a quantum computer. Phys. Rev. E 1999, 59, 2429– 2438, DOI: 10.1103/PhysRevE.59.2429
  There is no corresponding record for this reference.
14. 14
  Aspuru-Guzik, A.; Dutoi, A. D.; Love, P. J.; Head-Gordon, M. Simulated Quantum Computation of Molecular Energies. Science 2005, 309, 1704, DOI: 10.1126/science.1113479
  There is no corresponding record for this reference.
15. 15
  Peruzzo, A.; McClean, J.; Shadbolt, P.; Yung, M.-H.; Zhou, X.-Q.; Love, P. J.; Aspuru-Guzik, A.; O’Brien, J. L. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 2014, 5, 4213, DOI: 10.1038/ncomms5213
  There is no corresponding record for this reference.
16. 16
  McClean, J. R.; Romero, J.; Babbush, R.; Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 2016, 18, 023023 DOI: 10.1088/1367-2630/18/2/023023
  There is no corresponding record for this reference.
17. 17
  McArdle, S.; Endo, S.; Aspuru-Guzik, A.; Benjamin, S. C.; Yuan, X. Quantum computational chemistry. Rev. Mod. Phys. 2020, 92, 015003 DOI: 10.1103/RevModPhys.92.015003
  There is no corresponding record for this reference.
18. 18
  Tilly, J.; Chen, H.; Cao, S.; Picozzi, D.; Setia, K.; Li, Y.; Grant, E.; Wossnig, L.; Rungger, I.; Booth, G. H.; Tennyson, J. The Variational Quantum Eigensolver: A review of methods and best practices. Phys. Rep. 2022, 986, 1– 128, DOI: 10.1016/j.physrep.2022.08.003
  There is no corresponding record for this reference.
19. 19
  Kitaev, A. Y. Quantum measurements and the Abelian Stabilizer Problem. 1995; https://arxiv.org/abs/quant-ph/9511026 (accessed 2025-02-23).
  There is no corresponding record for this reference.
20. 20
  Cao, Y.; Romero, J.; Olson, J. P.; Degroote, M.; Johnson, P. D.; Kieferová, M.; Kivlichan, I. D.; Menke, T.; Peropadre, B.; Sawaya, N. P. D.; Sim, S.; Veis, L.; Aspuru-Guzik, A. Quantum Chemistry in the Age of Quantum Computing. Chem. Rev. 2019, 119, 10856– 10915, DOI: 10.1021/acs.chemrev.8b00803
  There is no corresponding record for this reference.
21. 21
  Lee, S. Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry. Nat. Commun. 2023, 14, 1952, DOI: 10.1038/s41467-023-37587-6
  There is no corresponding record for this reference.
22. 22
  Huggins, W. J.; McClean, J. R.; Rubin, N. C.; Jiang, Z.; Wiebe, N.; Whaley, K. B.; Babbush, R. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. npj Quantum Information 2021, 7, 23, DOI: 10.1038/s41534-020-00341-7
  There is no corresponding record for this reference.
23. 23
  Hashim, A. . A Practical Introduction to Benchmarking and Characterization of Quantum Computers. 2024; https://arxiv.org/abs/2408.12064 (accessed 2025-02-23).
  There is no corresponding record for this reference.
24. 24
  Chuang, I. L.; Nielsen, M. A. Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt. 1997, 44, 2455, DOI: 10.1080/09500349708231894
  There is no corresponding record for this reference.
25. 25
  Poyatos, J. F.; Cirac, J. I.; Zoller, P. Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate. Phys. Rev. Lett. 1997, 78, 390– 393, DOI: 10.1103/PhysRevLett.78.390
  There is no corresponding record for this reference.
26. 26
  Mohseni, M.; Rezakhani, A. T.; Lidar, D. A. Quantum-process tomography: Resource analysis of different strategies. Phys. Rev. A 2008, 77, 032322 DOI: 10.1103/PhysRevA.77.032322
  There is no corresponding record for this reference.
27. 27
  Emerson, J.; Alicki, R.; Życzkowski, K. Scalable noise estimation with random unitary operators. Journal of Optics B: Quantum and Semiclassical Optics 2005, 7, S347, DOI: 10.1088/1464-4266/7/10/021
  There is no corresponding record for this reference.
28. 28
  Knill, E.; Leibfried, D.; Reichle, R.; Britton, J.; Blakestad, R. B.; Jost, J. D.; Langer, C.; Ozeri, R.; Seidelin, S.; Wineland, D. J. Randomized benchmarking of quantum gates. Phys. Rev. A 2008, 77, 012307 DOI: 10.1103/PhysRevA.77.012307
  There is no corresponding record for this reference.
29. 29
  Magesan, E.; Gambetta, J. M.; Emerson, J. Scalable and Robust Randomized Benchmarking of Quantum Processes. Phys. Rev. Lett. 2011, 106, 180504, DOI: 10.1103/PhysRevLett.106.180504
  There is no corresponding record for this reference.
30. 30
  Helsen, J.; Roth, I.; Onorati, E.; Werner, A. H.; Eisert, J. General Framework for Randomized Benchmarking. PRX Quantum 2022, 3, 020357 DOI: 10.1103/PRXQuantum.3.020357
  There is no corresponding record for this reference.
31. 31
  Merkel, S. T.; Gambetta, J. M.; Smolin, J. A.; Poletto, S.; Córcoles, A. D.; Johnson, B. R.; Ryan, C. A.; Steffen, M. Self-consistent quantum process tomography. Phys. Rev. A 2013, 87, 062119 DOI: 10.1103/PhysRevA.87.062119
  There is no corresponding record for this reference.
32. 32
  Blume-Kohout, R.; Gamble, J. K.; Nielsen, E.; Mizrahi, J.; Sterk, J. D.; Maunz, P. Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit. 2013; https://arxiv.org/abs/1310.4492 (accessed 2025-02-23).
  There is no corresponding record for this reference.
33. 33
  Kim, D.; Ward, D. R.; Simmons, C. B.; Gamble, J. K.; Blume-Kohout, R.; Nielsen, E.; Savage, D. E.; Lagally, M. G.; Friesen, M.; Coppersmith, S. N.; Eriksson, M. A. Microwave-driven coherent operation of a semiconductor quantum dot charge qubit. Nat. Nanotechnol. 2015, 10, 243– 247, DOI: 10.1038/nnano.2014.336
  There is no corresponding record for this reference.
34. 34
  Bäumer, E.; Tripathi, V.; Seif, A.; Lidar, D.; Wang, D. S. Quantum Fourier Transform Using Dynamic Circuits. Phys. Rev. Lett. 2024, 133, 150602– 150609, DOI: 10.1103/PhysRevLett.133.150602
  There is no corresponding record for this reference.
35. 35
  da Silva, M. P.; Landon-Cardinal, O.; Poulin, D. Practical characterization of quantum devices without tomography. Phys. Rev. Lett. 2011, 107, 210404– 210409, DOI: 10.1103/PhysRevLett.107.210404
  There is no corresponding record for this reference.
36. 36
  Flammia, S. T.; Liu, Y.-K. Direct Fidelity Estimation from Few Pauli Measurements. Phys. Rev. Lett. 2011, 106, 230501, DOI: 10.1103/PhysRevLett.106.230501
  There is no corresponding record for this reference.
37. 37
  Neill, C. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 2018, 360, 195– 199, DOI: 10.1126/science.aao4309
  There is no corresponding record for this reference.
38. 38
  Boixo, S.; Isakov, S. V.; Smelyanskiy, V. N.; Babbush, R.; Ding, N.; Jiang, Z.; Bremner, M. J.; Martinis, J. M.; Neven, H. Characterizing quantum supremacy in near-term devices. Nature Physics 2018, 14, 595– 600
  There is no corresponding record for this reference.
39. 39
  Arute, F. Quantum supremacy using a programmable superconducting processor. Nature 2019, 574, 505– 510
  There is no corresponding record for this reference.
40. 40
  Kueng, R.; Long, D. M.; Doherty, A. C.; Flammia, S. T. Comparing Experiments to the Fault-Tolerance Threshold. Phys. Rev. Lett. 2016, 117, 170502, DOI: 10.1103/PhysRevLett.117.170502
  There is no corresponding record for this reference.
41. 41
  Barnes, J. P.; Trout, C. J.; Lucarelli, D.; Clader, B. D. Quantum error-correction failure distributions: Comparison of coherent and stochastic error models. Phys. Rev. A 2017, 95, 062338 DOI: 10.1103/PhysRevA.95.062338
  There is no corresponding record for this reference.
42. 42
  Ng, H. K.; Preskill, J. Fault-tolerant quantum computation versus Gaussian noise. Phys. Rev. A 2009, 79, 032318. DOI: 10.1103/PhysRevA.79.032318
  There is no corresponding record for this reference.
43. 43
  Wallman, J.; Granade, C.; Harper, R.; Flammia, S. T. Estimating the coherence of noise. New J. Phys. 2015, 17, 113020, DOI: 10.1088/1367-2630/17/11/113020
  There is no corresponding record for this reference.
44. 44
  Sanders, Y. R.; Wallman, J. J.; Sanders, B. C. Bounding quantum gate error rate based on reported average fidelity. New J. Phys. 2016, 18, 012002 DOI: 10.1088/1367-2630/18/1/012002
  There is no corresponding record for this reference.
45. 45
  Hashim, A.; Seritan, S.; Proctor, T.; Rudinger, K.; Goss, N.; Naik, R. K.; Kreikebaum, J. M.; Santiago, D. I.; Siddiqi, I. Benchmarking quantum logic operations relative to thresholds for fault tolerance. npj Quantum Information 2023, 9, 109, DOI: 10.1038/s41534-023-00764-y
  There is no corresponding record for this reference.
46. 46
  Wallman, J. J.; Emerson, J. Noise tailoring for scalable quantum computation via randomized compiling. Phys. Rev. A 2016, 94, 052325– 052334, DOI: 10.1103/PhysRevA.94.052325
  There is no corresponding record for this reference.
47. 47
  Hashim, A.; Naik, R. K.; Morvan, A.; Ville, J.-L.; Mitchell, B.; Kreikebaum, J. M.; Davis, M.; Smith, E.; Iancu, C.; O’Brien, K. P.; Hincks, I.; Wallman, J. J.; Emerson, J.; Siddiqi, I. Randomized Compiling for Scalable Quantum Computing on a Noisy Superconducting Quantum Processor. Phys. Rev. X 2021, 11, 041039 DOI: 10.1103/PhysRevX.11.041039
  There is no corresponding record for this reference.
48. 48
  van den Berg, E.; Minev, Z. K.; Kandala, A.; Temme, K. Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors. Nat. Phys. 2023, 19, 1116– 1121, DOI: 10.1038/s41567-023-02042-2
  There is no corresponding record for this reference.
49. 49
  Seif, A.; Liao, H.; Tripathi, V.; Krsulich, K.; Malekakhlagh, M.; Amico, M.; Jurcevic, P.; Javadi-Abhari, A. Suppressing Correlated Noise in Quantum Computers via Context-Aware Compiling. 51st Annual International Symposium on Computer Architecture (ISCA) 2024, 310– 324
  There is no corresponding record for this reference.
50. 50
  Gross, J. A.; Genois, E.; Debroy, D. M.; Zhang, Y.; Mruczkiewicz, W.; Cian, Z.-P.; Jiang, Z. Characterizing Coherent Errors using Matrix-Element Amplification. npj Quantum Information 2024, 10, 123, DOI: 10.1038/s41534-024-00917-7
  There is no corresponding record for this reference.
51. 51
  Bravyi, S.; Englbrecht, M.; König, R.; Peard, N. Correcting coherent errors with surface codes. npj Quantum Information 2018, 4, 55, DOI: 10.1038/s41534-018-0106-y
  There is no corresponding record for this reference.
52. 52
  Barenco, A.; Bennett, C. H.; Cleve, R.; DiVincenzo, D. P.; Margolus, N.; Shor, P.; Sleator, T.; Smolin, J. A.; Weinfurter, H. Elementary gates for quantum computation. Phys. Rev. A 1995, 52, 3457– 3467, DOI: 10.1103/PhysRevA.52.3457
  There is no corresponding record for this reference.
53. 53
  Erhard, A.; Wallman, J. J.; Postler, L.; Meth, M.; Stricker, R.; Martinez, E. A.; Schindler, P.; Monz, T.; Emerson, J.; Blatt, R. Characterizing large-scale quantum computers via cycle benchmarking. Nat. Commun. 2019, 10, 5347, DOI: 10.1038/s41467-019-13068-7
  There is no corresponding record for this reference.
54. 54
  McKay, D. C.; Wood, C. J.; Sheldon, S.; Chow, J. M.; Gambetta, J. M. Efficient Z gates for quantum computing. Phys. Rev. A 2017, 96, 022330 DOI: 10.1103/PhysRevA.96.022330
  There is no corresponding record for this reference.
55. 55
  Cross, A. W.; Bishop, L. S.; Sheldon, S.; Nation, P. D.; Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 2019, 100, 032328 DOI: 10.1103/PhysRevA.100.032328
  There is no corresponding record for this reference.
56. 56
  Dankert, C.; Cleve, R.; Emerson, J.; Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 2009, 80, 012304 DOI: 10.1103/PhysRevA.80.012304
  There is no corresponding record for this reference.
57. 57
  Proctor, T.; Rudinger, K.; Young, K.; Nielsen, E.; Blume-Kohout, R. Measuring the capabilities of quantum computers. Nat. Phys. 2022, 18, 75– 79, DOI: 10.1038/s41567-021-01409-7
  There is no corresponding record for this reference.
58. 58
  Cincio, L.; Subası, Y.; Sornborger, A. T; Coles, P. J Learning the quantum algorithm for state overlap. New J. Phys. 2018, 20, 113022, DOI: 10.1088/1367-2630/aae94a
  There is no corresponding record for this reference.
59. 59
  Reich, D. M.; Gualdi, G.; Koch, C. P. Optimal Strategies for Estimating the Average Fidelity of Quantum Gates. Phys. Rev. Lett. 2013, 111, 200401– 200406, DOI: 10.1103/PhysRevLett.111.200401
  There is no corresponding record for this reference.
60. 60
  Torlai, G.; Mazzola, G.; Carrasquilla, J.; Troyer, M.; Melko, R.; Carleo, G. Neural-network quantum state tomography. Nat. Phys. 2018, 14, 447– 450, DOI: 10.1038/s41567-018-0048-5
  There is no corresponding record for this reference.
61. 61
  Koutny, D.; Motka, L.; Hradil, Z.; Rehacek, J.; Sanchez-Soto, L. L. Neural-network quantum state tomography. Phys. Rev. A 2022, 106. DOI: 10.1103/PhysRevA.106.012409
  There is no corresponding record for this reference.
62. 62
  Gu, Y.; Zhuang, W.-F.; Chai, X.; Liu, D. E. Benchmarking universal quantum gates via channel spectrum. Nat. Commun. 2023, 14, 5880, DOI: 10.1038/s41467-023-41598-8
  There is no corresponding record for this reference.
63. 63
  McKay, D. C.; Hincks, I.; Pritchett, E. J.; Carroll, M.; Govia, L. C. G.; Merkel, S. T. Benchmarking Quantum Processor Performance at Scale. 2023; https://arxiv.org/abs/2311.05933 (accessed: 2025-02-23).
  There is no corresponding record for this reference.
64. 64
  Breuer, H.-P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press: Oxford, 2002.
  There is no corresponding record for this reference.
65. 65
  Kraus, K. States, Effects and Operations; Fundamental Notions of Quantum Theory; Academic, 1983.
  There is no corresponding record for this reference.
66. 66
  Nielsen, M. A.; Chuang, I. L. Quantum computation and quantum information; Cambridge University Press, 2010.
  There is no corresponding record for this reference.
67. 67
  Jamiołkowski, A. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics 1972, 3, 275– 278, DOI: 10.1016/0034-4877(72)90011-0
  There is no corresponding record for this reference.
68. 68
  Choi, M.-D. Completely positive linear maps on complex matrices. Linear Algebra and its Applications 1975, 10, 285– 290, DOI: 10.1016/0024-3795(75)90075-0
  There is no corresponding record for this reference.
69. 69
  Uhlmann, A. The “transition probability”in the state space of a ∗-algebra. Reports on Mathematical Physics 1976, 9, 273– 279, DOI: 10.1016/0034-4877(76)90060-4
  There is no corresponding record for this reference.
70. 70
  Knill, E.; Laflamme, R. Theory of quantum error-correcting codes. Phys. Rev. A 1997, 55, 900– 911, DOI: 10.1103/PhysRevA.55.900
  There is no corresponding record for this reference.
71. 71
  Tripathi, V.; Chen, H.; Levenson-Falk, E.; Lidar, D. A. Modeling Low- and High-Frequency Noise in Transmon Qubits with Resource-Efficient Measurement. PRX Quantum 2024, 5, 010320 DOI: 10.1103/PRXQuantum.5.010320
  There is no corresponding record for this reference.
72. 72
  Byrd, M. S.; Lidar, D. A.; Wu, L.-A.; Zanardi, P. Universal leakage elimination. Phys. Rev. A 2005, 71, 052301 DOI: 10.1103/PhysRevA.71.052301
  There is no corresponding record for this reference.
73. 73
  Koch, J.; Yu, T. M.; Gambetta, J.; Houck, A. A.; Schuster, D. I.; Majer, J.; Blais, A.; Devoret, M. H.; Girvin, S. M.; Schoelkopf, R. J. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 2007, 76, 042319 DOI: 10.1103/PhysRevA.76.042319
  There is no corresponding record for this reference.
74. 74
  Wu, L.-A.; Walther, P.; Lidar, D. A. No-go theorem for passive single-rail linear optical quantum computing. Sci. Rep. 2013, 3, 1394, DOI: 10.1038/srep01394
  There is no corresponding record for this reference.
75. 75
  Levine, H. Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons. Physical Review X 2024, 14, 011051– 011072, DOI: 10.1103/PhysRevX.14.011051
  There is no corresponding record for this reference.
76. 76
  Grünzweig, T.; Hilliard, A.; McGovern, M.; Andersen, M. F. Near-deterministic preparation of a single atom in an optical microtrap. Nat. Phys. 2010, 6, 951– 954, DOI: 10.1038/nphys1778
  There is no corresponding record for this reference.
77. 77
  Wu, Y.; Kolkowitz, S.; Puri, S.; Thompson, J. D. Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays. Nat. Commun. 2022, 13, 4657, DOI: 10.1038/s41467-022-32094-6
  There is no corresponding record for this reference.
78. 78
  Varbanov, B. M.; Battistel, F.; Tarasinski, B. M.; Ostroukh, V. P.; O’Brien, T. E.; DiCarlo, L.; Terhal, B. M. Leakage detection for a transmon-based surface code. npj Quantum Information 2020, 6, 102, DOI: 10.1038/s41534-020-00330-w
  There is no corresponding record for this reference.
79. 79
  McEwen, M. Removing leakage-induced correlated errors in superconducting quantum error correction. Nat. Commun. 2021, 12, 1761, DOI: 10.1038/s41467-021-21982-y
  There is no corresponding record for this reference.
80. 80
  Gourlay, I.; Snowdon, J. F. Concatenated Coding in the Presence of Dephasing. Phys. Rev. A 2000, 62, 022308 DOI: 10.1103/PhysRevA.62.022308
  There is no corresponding record for this reference.
81. 81
  Evans, Z. W. E.; Stephens, A. M.; Cole, J. H.; Hollenberg, L. C. L. Error Correction Optimisation in the Presence of X/Z Asymmetry. 2007.
  There is no corresponding record for this reference.
82. 82
  Stephens, A. M.; Evans, Z. W. E.; Devitt, S. J.; Hollenberg, L. C. L. Asymmetric Quantum Error Correction via Code Conversion. Phys. Rev. A 2008, 77, 062335 DOI: 10.1103/PhysRevA.77.062335
  There is no corresponding record for this reference.
83. 83
  Aliferis, P.; Preskill, J. Fault-tolerant quantum computation against biased noise. Phys. Rev. A 2008, 78, 052331 DOI: 10.1103/PhysRevA.78.052331
  There is no corresponding record for this reference.
84. 84
  Stephens, A. M.; Munro, W. J.; Nemoto, K. High-Threshold Topological Quantum Error Correction against Biased Noise. Phys. Rev. A 2013, 88, 060301 DOI: 10.1103/PhysRevA.88.060301
  There is no corresponding record for this reference.
85. 85
  Brooks, P.; Preskill, J. Fault-Tolerant Quantum Computation with Asymmetric Bacon-Shor Codes. Phys. Rev. A 2013, 87, 032310 DOI: 10.1103/PhysRevA.87.032310
  There is no corresponding record for this reference.
86. 86
  Bonilla Ataides, J. P.; Tuckett, D. K.; Bartlett, S. D.; Flammia, S. T.; Brown, B. J. The XZZX surface code. Nat. Commun. 2021, 12, 2172, DOI: 10.1038/s41467-021-22274-1
  There is no corresponding record for this reference.
87. 87
  Chruscinski, D.; Kossakowski, A. Non-Markovian Quantum Dynamics: Local versus Nonlocal. Phys. Rev. Lett. 2010, 104, 070406 DOI: 10.1103/PhysRevLett.104.070406
  There is no corresponding record for this reference.
88. 88
  Rivas, Á.; Huelga, S. F.; Plenio, M. B. Quantum non-Markovianity: characterization, quantification and detection. Rep. Prog. Phys. 2014, 77, 094001 DOI: 10.1088/0034-4885/77/9/094001
  There is no corresponding record for this reference.
89. 89
  Shabani, A.; Lidar, D. A. Completely positive post-Markovian master equation via a measurement approach. Phys. Rev. A 2005, 71, 020101 DOI: 10.1103/PhysRevA.71.020101
  There is no corresponding record for this reference.
90. 90
  Zhang, H.; Pokharel, B.; Levenson-Falk, E.; Lidar, D. Predicting Non-Markovian Superconducting-Qubit Dynamics from Tomographic Reconstruction. Physical Review Applied 2022, 17, 054018 DOI: 10.1103/PhysRevApplied.17.054018
  There is no corresponding record for this reference.
91. 91
  Zhang, J.; Vala, J.; Sastry, S.; Whaley, K. B. Geometric theory of nonlocal two-qubit operations. Phys. Rev. A 2003, 67, 042313 DOI: 10.1103/PhysRevA.67.042313
  There is no corresponding record for this reference.
92. 92
  Sete, E. A.; Didier, N.; Chen, A. Q.; Kulshreshtha, S.; Manenti, R.; Poletto, S. Parametric-Resonance Entangling Gates with a Tunable Coupler. Phys. Rev. Appl. 2021, 16, 024050 DOI: 10.1103/PhysRevApplied.16.024050
  There is no corresponding record for this reference.
93. 93
  Sete, E. A.; Tripathi, V.; Valery, J. A.; Lidar, D.; Mutus, J. Y. Error Budget of a Parametric Resonance Entangling Gate with a Tunable Coupler. Physical Review Applied 2024, 22, 014059 DOI: 10.1103/PhysRevApplied.22.014059
  There is no corresponding record for this reference.
94. 94
  Tripathi, V.; Khezri, M.; Korotkov, A. N. Operation and intrinsic error budget of a two-qubit cross-resonance gate. Phys. Rev. A 2019, 100, 012301 DOI: 10.1103/PhysRevA.100.012301
  There is no corresponding record for this reference.
95. 95
  Gaikwad, C.; Kowsari, D.; Brame, C.; Song, X.; Zhang, H.; Esposito, M.; Ranadive, A.; Cappelli, G.; Roch, N.; Levenson-Falk, E. M.; Murch, K. W. Entanglement Assisted Probe of the Non-Markovian to Markovian Transition in Open Quantum System Dynamics. Phys. Rev. Lett. 2024, 132, 200401, DOI: 10.1103/PhysRevLett.132.200401
  There is no corresponding record for this reference.
96. 96
  Dubi, Y.; Di Ventra, M. Relaxation times in an open interacting two-qubit system. Phys. Rev. A 2009, 79, 012328 DOI: 10.1103/PhysRevA.79.012328
  There is no corresponding record for this reference.
97. 97
  Greenbaum, D. Introduction to Quantum Gate Set Tomography. 2015, arXiv:1509.02921 [quant-ph] (accessed: 2025-02-23).
  There is no corresponding record for this reference.
98. 98
  Brown, W. G.; Eastin, B. Randomized Benchmarking with Restricted Gate Sets. Phys. Rev. A 2018, 97, 062323 DOI: 10.1103/PhysRevA.97.062323
  There is no corresponding record for this reference.
99. 99
  Seif, A.; Chen, S.; Majumder, S.; Liao, H.; Wang, D. S.; Malekakhlagh, M.; Javadi-Abhari, A.; Jiang, L.; Minev, Z. K. Entanglement-enhanced learning of quantum processes at scale. 2024; https://arxiv.org/abs/2408.03376 (accessed: 2025-02-23).
  There is no corresponding record for this reference.
100. 100
  Khan, M. Q.; Dong, W.; Norris, L. M.; Viola, L. Multiaxis quantum noise spectroscopy robust to errors in state preparation and measurement. Physical Review Applied 2024, 22, 024074 DOI: 10.1103/PhysRevApplied.22.024074
  There is no corresponding record for this reference.
101. 101
  Merkel, S. T.; Gambetta, J. M.; Smolin, J. A.; Poletto, S.; Córcoles, A. D.; Johnson, B. R.; Ryan, C. A.; Steffen, M. Self-consistent quantum process tomography. Phys. Rev. A 2013, 87, 062119 DOI: 10.1103/PhysRevA.87.062119
  There is no corresponding record for this reference.
102. 102
  Magesan, E.; Gambetta, J. M.; Emerson, J. Characterizing quantum gates via randomized benchmarking. Phys. Rev. A 2012, 85, 042311 DOI: 10.1103/PhysRevA.85.042311
  There is no corresponding record for this reference.
103. 103
  Aaronson, S.; Gottesman, D. Improved simulation of stabilizer circuits. Phys. Rev. A 2004, 70, 052328 DOI: 10.1103/PhysRevA.70.052328
  There is no corresponding record for this reference.
104. 104
  Nielsen, M. A. A simple formula for the average gate fidelity of a quantum dynamical operation. Phys. Lett. A 2002, 303, 249– 252, DOI: 10.1016/S0375-9601(02)01272-0
  There is no corresponding record for this reference.
105. 105
  Viola, L.; Knill, E. Random Decoupling Schemes for Quantum Dynamical Control and Error Suppression. Phys. Rev. Lett. 2005, 94, 060502 DOI: 10.1103/PhysRevLett.94.060502
  There is no corresponding record for this reference.
106. 106
  Santos, L. F.; Viola, L. Dynamical control of qubit coherence: Random versus deterministic schemes. Phys. Rev. A 2005, 72, 062303 DOI: 10.1103/PhysRevA.72.062303
  There is no corresponding record for this reference.
107. 107
  Santos, L. F.; Viola, L. Enhanced Convergence and Robust Performance of Randomized Dynamical Decoupling. Phys. Rev. Lett. 2006, 97, 150501, DOI: 10.1103/PhysRevLett.97.150501
  There is no corresponding record for this reference.
108. 108
  Magesan, E.; Gambetta, J. M.; Johnson, B. R.; Ryan, C. A.; Chow, J. M.; Merkel, S. T.; da Silva, M. P.; Keefe, G. A.; Rothwell, M. B.; Ohki, T. A.; Ketchen, M. B.; Steffen, M. Efficient Measurement of Quantum Gate Error by Interleaved Randomized Benchmarking. Phys. Rev. Lett. 2012, 109, 080505 DOI: 10.1103/PhysRevLett.109.080505
  There is no corresponding record for this reference.
109. 109
  Gambetta, J. M.; Córcoles, A. D.; Merkel, S. T.; Johnson, B. R.; Smolin, J. A.; Chow, J. M.; Ryan, C. A.; Rigetti, C.; Poletto, S.; Ohki, T. A.; Ketchen, M. B.; Steffen, M. Characterization of Addressability by Simultaneous Randomized Benchmarking. Phys. Rev. Lett. 2012, 109, 240504, DOI: 10.1103/PhysRevLett.109.240504
  There is no corresponding record for this reference.
110. 110
  Wallman, J. J.; Barnhill, M.; Emerson, J. Robust characterization of leakage errors. New J. Phys. 2016, 18, 043021 DOI: 10.1088/1367-2630/18/4/043021
  There is no corresponding record for this reference.
111. 111
  Wood, C. J.; Gambetta, J. M. Quantification and characterization of leakage errors. Phys. Rev. A 2018, 97, 032306 DOI: 10.1103/PhysRevA.97.032306
  There is no corresponding record for this reference.
112. 112
  Fogarty, M. A.; Veldhorst, M.; Harper, R.; Yang, C. H.; Bartlett, S. D.; Flammia, S. T.; Dzurak, A. S. Nonexponential fidelity decay in randomized benchmarking with low-frequency noise. Phys. Rev. A 2015, 92, 022326 DOI: 10.1103/PhysRevA.92.022326
  There is no corresponding record for this reference.
113. 113
  Wallman, J. J. Randomized benchmarking with gate-dependent noise. Quantum 2018, 2, 47, DOI: 10.22331/q-2018-01-29-47
  There is no corresponding record for this reference.
114. 114
  Proctor, T.; Seritan, S.; Rudinger, K.; Nielsen, E.; Blume-Kohout, R.; Young, K. Scalable Randomized Benchmarking of Quantum Computers Using Mirror Circuits. Phys. Rev. Lett. 2022, 129, 150502, DOI: 10.1103/PhysRevLett.129.150502
  There is no corresponding record for this reference.
115. 115
  Helsen, J.; Xue, X.; Vandersypen, L. M. K.; Wehner, S. A new class of efficient randomized benchmarking protocols. npj Quantum Information 2019, 5, 71, DOI: 10.1038/s41534-019-0182-7
  There is no corresponding record for this reference.
116. 116
  Helsen, J.; Roth, I.; Onorati, E.; Werner, A.; Eisert, J. General Framework for Randomized Benchmarking. PRX Quantum 2022, 3, 020357 DOI: 10.1103/PRXQuantum.3.020357
  There is no corresponding record for this reference.
117. 117
  Arrighi, P.; Patricot, C. On quantum operations as quantum states. Annals of Physics 2004, 311, 26– 52, DOI: 10.1016/j.aop.2003.11.005
  There is no corresponding record for this reference.
118. 118
  D’Ariano, G. M.; Lo Presti, P. Quantum Tomography for Measuring Experimentally the Matrix Elements of an Arbitrary Quantum Operation. Phys. Rev. Lett. 2001, 86, 4195– 4198, DOI: 10.1103/PhysRevLett.86.4195
  There is no corresponding record for this reference.
119. 119
  Leung, D. W. Choi’s proof as a recipe for quantum process tomography. Journal of Mathematical Physics 2003, 44, 528, DOI: 10.1063/1.1518554
  There is no corresponding record for this reference.
120. 120
  Altepeter, J. B.; Branning, D.; Jeffrey, E.; Wei, T. C.; Kwiat, P. G.; Thew, R. T.; O’Brien, J. L.; Nielsen, M. A.; White, A. G. Ancilla-Assisted Quantum Process Tomography. Phys. Rev. Lett. 2003, 90, 193601, DOI: 10.1103/PhysRevLett.90.193601
  There is no corresponding record for this reference.
121. 121
  D’Ariano, G. M.; Lo Presti, P. Imprinting Complete Information about a Quantum Channel on its Output State. Phys. Rev. Lett. 2003, 91, 047902 DOI: 10.1103/PhysRevLett.91.047902
  There is no corresponding record for this reference.
122. 122
  Mohseni, M.; Lidar, D. A. Direct Characterization of Quantum Dynamics. Phys. Rev. Lett. 2006, 97, 170501, DOI: 10.1103/PhysRevLett.97.170501
  There is no corresponding record for this reference.
123. 123
  Mohseni, M.; Lidar, D. A. Direct characterization of quantum dynamics: General theory. Phys. Rev. A 2007, 75, 062331 DOI: 10.1103/PhysRevA.75.062331
  There is no corresponding record for this reference.
124. 124
  Mohseni, M.; Rezakhani, A. T.; Aspuru-Guzik, A. Direct estimation of single- and two-qubit Hamiltonians and relaxation rates. Phys. Rev. A 2008, 77, 042320. DOI: 10.1103/PhysRevA.77.042320
  There is no corresponding record for this reference.
125. 125
  Graham, T. M.; Barreiro, J. T.; Mohseni, M.; Kwiat, P. G. Hyperentanglement-Enabled Direct Characterization of Quantum Dynamics. Phys. Rev. Lett. 2013, 110, 060404 DOI: 10.1103/PhysRevLett.110.060404
  There is no corresponding record for this reference.
126. 126
  Nigg, D.; Barreiro, J. T.; Schindler, P.; Mohseni, M.; Monz, T.; Chwalla, M.; Hennrich, M.; Blatt, R. Experimental Characterization of Quantum Dynamics Through Many-Body Interactions. Phys. Rev. Lett. 2013, 110, 060403 DOI: 10.1103/PhysRevLett.110.060403
  There is no corresponding record for this reference.
127. 127
  Gottesman, D. Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 1996, 54, 1862, DOI: 10.1103/PhysRevA.54.1862
  There is no corresponding record for this reference.
128. 128
  Blume-Kohout, R.; Gamble, J. K.; Nielsen, E.; Rudinger, K.; Mizrahi, J.; Fortier, K.; Maunz, P. Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography. Nat. Commun. 2017, 8, 14485, DOI: 10.1038/ncomms14485
  There is no corresponding record for this reference.
129. 129
  Rol, M. A.; Bultink, C. C.; O’Brien, T. E.; de Jong, S. R.; Theis, L. S.; Fu, X.; Luthi, F.; Vermeulen, R. F. L.; de Sterke, J. C.; Bruno, A.; Deurloo, D.; Schouten, R. N.; Wilhelm, F. K.; DiCarlo, L. Restless Tuneup of High-Fidelity Qubit Gates. Phys. Rev. Appl. 2017, 7, 041001 DOI: 10.1103/PhysRevApplied.7.041001
  There is no corresponding record for this reference.
130. 130
  Mavadia, S.; Edmunds, C. L.; Hempel, C.; Ball, H.; Roy, F.; Stace, T. M.; Biercuk, M. J. Experimental quantum verification in the presence of temporally correlated noise. npj Quantum Information 2018, 4, 7, DOI: 10.1038/s41534-017-0052-0
  There is no corresponding record for this reference.
131. 131
  Hempel, C.; Maier, C.; Romero, J.; McClean, J.; Monz, T.; Shen, H.; Jurcevic, P.; Lanyon, B. P.; Love, P.; Babbush, R.; Aspuru-Guzik, A.; Blatt, R.; Roos, C. F. Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator. Phys. Rev. X 2018, 8, 031022 DOI: 10.1103/PhysRevX.8.031022
  There is no corresponding record for this reference.
132. 132
  Song, C.; Cui, J.; Wang, H.; Hao, J.; Feng, H.; Li, Y. Quantum computation with universal error mitigation on a superconducting quantum processor. Science Advances 2019, 5, eaaw5686 DOI: 10.1126/sciadv.aaw5686
  There is no corresponding record for this reference.
133. 133
  Cincio, L.; Rudinger, K.; Sarovar, M.; Coles, P. J. Machine Learning of Noise-Resilient Quantum Circuits. PRX Quantum 2021, 2, 010324 DOI: 10.1103/PRXQuantum.2.010324
  There is no corresponding record for this reference.
134. 134
  Hong, S. S.; Papageorge, A. T.; Sivarajah, P.; Crossman, G.; Didier, N.; Polloreno, A. M.; Sete, E. A.; Turkowski, S. W.; da Silva, M. P.; Johnson, B. R. Demonstration of a parametrically activated entangling gate protected from flux noise. Phys. Rev. A 2020, 101, 012302 DOI: 10.1103/PhysRevA.101.012302
  There is no corresponding record for this reference.
135. 135
  Joshi, M. K.; Elben, A.; Vermersch, B.; Brydges, T.; Maier, C.; Zoller, P.; Blatt, R.; Roos, C. F. Quantum Information Scrambling in a Trapped-Ion Quantum Simulator with Tunable Range Interactions. Phys. Rev. Lett. 2020, 124, 240505, DOI: 10.1103/PhysRevLett.124.240505
  There is no corresponding record for this reference.
136. 136
  Zhang, S.; Lu, Y.; Zhang, K.; Chen, W.; Li, Y.; Zhang, J.-N.; Kim, K. Error-mitigated quantum gates exceeding physical fidelities in a trapped-ion system. Nat. Commun. 2020, 11, 587, DOI: 10.1038/s41467-020-14376-z
  There is no corresponding record for this reference.
137. 137
  Ware, M.; Ribeill, G.; Ristè, D.; Ryan, C. A.; Johnson, B.; da Silva, M. P. Experimental Pauli-frame randomization on a superconducting qubit. Phys. Rev. A 2021, 103, 042604 DOI: 10.1103/PhysRevA.103.042604
  There is no corresponding record for this reference.
138. 138
  Dehollain, J. P.; Muhonen, J. T.; Blume-Kohout, R.; Rudinger, K. M.; Gamble, J. K.; Nielsen, E.; Laucht, A.; Simmons, S.; Kalra, R.; Dzurak, A. S.; Morello, A. Optimization of a solid-state electron spin qubit using gate set tomography. New J. Phys. 2016, 18, 103018, DOI: 10.1088/1367-2630/18/10/103018
  There is no corresponding record for this reference.
139. 139
  Kitaev, A.; Shen, A.; Vyalyi, M. Classical and Quantum Computation; Graduate studies in mathematics; American Mathematical Society, 2002.
  There is no corresponding record for this reference.
140. 140
  Scott, A. J. Tight informationally complete quantum measurements. Journal of Physics A: Mathematical and General 2006, 39, 13507, DOI: 10.1088/0305-4470/39/43/009
  There is no corresponding record for this reference.
141. 141
  Nielsen, E.; Gamble, J. K.; Rudinger, K.; Scholten, T.; Young, K.; Blume-Kohout, R. Gate Set Tomography. Quantum 2021, 5, 557, DOI: 10.22331/q-2021-10-05-557
  There is no corresponding record for this reference.
142. 142
  Householder, A. The Theory of Matrices in Numerical Analysis; Dover Books on Mathematics; Dover Publications, 2013.
  There is no corresponding record for this reference.
143. 143
  Nielsen, E.; Rudinger, K.; Proctor, T.; Russo, A.; Young, K.; Blume-Kohout, R. Probing quantum processor performance with pyGSTi. Quantum Science and Technology 2020, 5, 044002 DOI: 10.1088/2058-9565/ab8aa4
  There is no corresponding record for this reference.
144. 144
  Proctor, T.; Revelle, M.; Nielsen, E.; Rudinger, K.; Lobser, D.; Maunz, P.; Blume-Kohout, R.; Young, K. Detecting and tracking drift in quantum information processors. Nat. Commun. 2020, 11, 5396, DOI: 10.1038/s41467-020-19074-4
  There is no corresponding record for this reference.
145. 145
  Rudinger, K. M.; Ostrove, C. I.; Seritan, S. K.; Grace, M. D.; Nielsen, E.; Blume-Kohout, R. J.; Young, K. C. Two-Qubit Gate Set Tomography with Fewer Circuits. 2023; https://arxiv.org/abs/2307.15767 (accessed: 2025-02-23).
  There is no corresponding record for this reference.
146. 146
  Horodecki, M.; Horodecki, P.; Horodecki, R. General teleportation channel, singlet fraction, and quasidistillation. Phys. Rev. A 1999, 60, 1888– 1898, DOI: 10.1103/PhysRevA.60.1888
  There is no corresponding record for this reference.
147. 147
  Lu, D.; Li, H.; Trottier, D.-A.; Li, J.; Brodutch, A.; Krismanich, A. P.; Ghavami, A.; Dmitrienko, G. I.; Long, G.; Baugh, J.; Laflamme, R. Experimental Estimation of Average Fidelity of a Clifford Gate on a 7-qubit Quantum Processor. Phys. Rev. Lett. 2015, 114, 140505, DOI: 10.1103/PhysRevLett.114.140505
  There is no corresponding record for this reference.
148. 148
  Fedorov, A.; Steffen, L.; Baur, M.; da Silva, M. P.; Wallraff, A. Implementation of a Toffoli Gate with Superconducting Circuits. Nature 2012, 481, 170– 172, DOI: 10.1038/nature10713
  There is no corresponding record for this reference.
149. 149
  Wallman, J. J.; Flammia, S. T. Randomized benchmarking with confidence. New J. Phys. 2014, 16, 103032, DOI: 10.1088/1367-2630/16/10/103032
  There is no corresponding record for this reference.
150. 150
  Roth, I.; Kueng, R.; Kimmel, S.; Liu, Y.-K.; Gross, D.; Eisert, J.; Kliesch, M. Recovering quantum gates from few average gate fidelities. Phys. Rev. Lett. 2018, 121, 170502, DOI: 10.1103/PhysRevLett.121.170502
  There is no corresponding record for this reference.
151. 151
  Kalev, A.; Kyrillidis, A.; Linke, N. M. Validating and certifying stabilizer states. Phys. Rev. A 2019, 99, 042337 DOI: 10.1103/PhysRevA.99.042337
  There is no corresponding record for this reference.
152. 152
  Seshadri, A.; Ringbauer, M.; Spainhour, J.; Monz, T.; Becker, S. Theory of versatile fidelity estimation with confidence. Phys. Rev. A 2024, 110, 012431 DOI: 10.1103/PhysRevA.110.012431
  There is no corresponding record for this reference.
153. 153
  Seshadri, A.; Ringbauer, M.; Spainhour, J.; Blatt, R.; Monz, T.; Becker, S. Versatile Fidelity Estimation with Confidence. Phys. Rev. Lett. 2024, 133, 020402 DOI: 10.1103/PhysRevLett.133.020402
  There is no corresponding record for this reference.
154. 154
  Kosut, R. L.; Rabitz, H.; Grace, M. D. Adaptive quantum control via direct fidelity estimation and indirect model-based parametric process tomography. 52nd IEEE Conference on Decision and Control. Firenze, 2013; pp 1247– 1252.
  There is no corresponding record for this reference.
155. 155
  Lu, Y.; Sim, J. Y.; Suzuki, J.; Englert, B.-G.; Ng, H. K. Direct estimation of minimum gate fidelity. Phys. Rev. A 2020, 102, 022410 DOI: 10.1103/PhysRevA.102.022410
  There is no corresponding record for this reference.
156. 156
  Liu, Y.; Otten, M.; Bassirianjahromi, R.; Jiang, L.; Fefferman, B. Benchmarking near-term quantum computers via random circuit sampling. 2022; http://arxiv.org/abs/2105.05232, arXiv: 2105.05232 [quant-ph] (accessed: 2025-02-23).
  There is no corresponding record for this reference.
157. 157
  Elements of Information Theory; John Wiley & Sons, Ltd, 2005; Chapter 4, pp 71– 101.
  There is no corresponding record for this reference.
158. 158
  Ledoux, M. The Concentration of Measure Phenomenon; American Mathematical Soc., 2001; Google-Books-ID: rRKChz4Om6gC.
  There is no corresponding record for this reference.
159. 159
  Boixo, S.; Smelyanskiy, V. N.; Neven, H. Fourier analysis of sampling from noisy chaotic quantum circuits. 2017; http://arxiv.org/abs/1708.01875, arXiv: 1708.01875 [quant-ph] (accessed: 2025-02-23).
  There is no corresponding record for this reference.
160. 160
  Choi, J.; Shaw, A. L.; Madjarov, I. S.; Xie, X.; Finkelstein, R.; Covey, J. P.; Cotler, J. S.; Mark, D. K.; Huang, H.-Y.; Kale, A.; Pichler, H.; Brandão, F. G. S. L.; Choi, S.; Endres, M. Preparing random states and benchmarking with many-body quantum chaos. Nature 2023, 613, 468– 473, DOI: 10.1038/s41586-022-05442-1
  There is no corresponding record for this reference.
161. 161
  Rinott, Y.; Shoham, T.; Kalai, G. Statistical Aspects of the Quantum Supremacy Demonstration. Statistical Science 2022, 37, 322– 347
  There is no corresponding record for this reference.
162. 162
  Dalzell, A. M.; Hunter-Jones, N.; Brandão, F. G. S. L. Random Quantum Circuits Anticoncentrate in Log Depth. PRX Quantum 2022, 3, 010333
  There is no corresponding record for this reference.
163. 163
  Huang, C.; Zhang, F.; Newman, M.; Cai, J.; Gao, X.; Tian, Z.; Wu, J.; Xu, H.; Yu, H.; Yuan, B.; Szegedy, M.; Shi, Y.; Chen, J. Classical Simulation of Quantum Supremacy Circuits. arXiv 2020 (accessed 2025-02-23). DOI: 10.48550/arXiv.2005.06787
  There is no corresponding record for this reference.
164. 164
  Zlokapa, A.; Villalonga, B.; Boixo, S.; Lidar, D. A. Boundaries of quantum supremacy via random circuit sampling. npj Quantum Information 2023, 9, 36, DOI: 10.1038/s41534-023-00703-x
  There is no corresponding record for this reference.
165. 165
  Kechedzhi, K.; Isakov, S. V.; Mandrà, S.; Villalonga, B.; Mi, X.; Boixo, S.; Smelyanskiy, V. Effective quantum volume, fidelity and computational cost of noisy quantum processing experiments. Future Generation Computer Systems 2024, 153, 431– 441, DOI: 10.1016/j.future.2023.12.002
  There is no corresponding record for this reference.
166. 166
  Arute, F. Quantum supremacy using a programmable superconducting processor. Nature 2019, 574, 505– 510, DOI: 10.1038/s41586-019-1666-5
  There is no corresponding record for this reference.
167. 167
  Harrow, A. W.; Mehraban, S. Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates. Communications in Mathematical Physics 2023, 401, 1531– 1626, DOI: 10.1007/s00220-023-04675-z
  There is no corresponding record for this reference.
168. 168
  Haferkamp, J.; Montealegre-Mora, F.; Heinrich, M.; Eisert, J.; Gross, D.; Roth, I. Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates. Communications in Mathematical Physics 2023, 397, 995– 1041, DOI: 10.1007/s00220-022-04507-6
  There is no corresponding record for this reference.
169. 169
  Gao, X.; Kalinowski, M.; Chou, C.-N.; Lukin, M. D.; Barak, B.; Choi, S. Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage. PRX Quantum 2024, 5, 010334 DOI: 10.1103/PRXQuantum.5.010334
  There is no corresponding record for this reference.
170. 170
  Tripathi, V.; Kowsari, D.; Saurav, K.; Zhang, H.; M.Levenson-Falk, E.; Lidar, D. A. Benchmarking quantum gates and circuits. 2024; https://arxiv.org/abs/2407.09942 (accessed: 2025-02-23).
  There is no corresponding record for this reference.
171. 171
  Burnett, J. J.; Bengtsson, A.; Scigliuzzo, M.; Niepce, D.; Kudra, M.; Delsing, P.; Bylander, J. Decoherence benchmarking of superconducting qubits. npj Quantum Information 2019, 5, 54, DOI: 10.1038/s41534-019-0168-5
  There is no corresponding record for this reference.
172. 172
  Hahn, E. L. Spin Echoes. Phys. Rev. 1950, 80, 580– 594, DOI: 10.1103/PhysRev.80.580
  There is no corresponding record for this reference.
173. 173
  Carr, H. Y.; Purcell, E. M. Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments. Phys. Rev. 1954, 94, 630– 638, DOI: 10.1103/PhysRev.94.630
  There is no corresponding record for this reference.
174. 174
  Meiboom, S.; Gill, D. Modified Spin?Echo Method for Measuring Nuclear Relaxation Times. Rev. Sci. Instrum. 1958, 29, 688– 691, DOI: 10.1063/1.1716296
  There is no corresponding record for this reference.
175. 175
  Viola, L.; Lloyd, S. Dynamical suppression of decoherence in two-state quantum systems. Phys. Rev. A 1998, 58, 2733– 2744, DOI: 10.1103/PhysRevA.58.2733
  There is no corresponding record for this reference.
176. 176
  Uhrig, G. S. Keeping a Quantum Bit Alive by Optimized π-Pulse Sequences. Phys. Rev. Lett. 2007, 98, 100504, DOI: 10.1103/PhysRevLett.98.100504
  There is no corresponding record for this reference.
177. 177
  Suter, D.; Álvarez, G. A. Colloquium: Protecting quantum information against environmental noise. Rev. Mod. Phys. 2016, 88, 041001 DOI: 10.1103/RevModPhys.88.041001
  There is no corresponding record for this reference.
178. 178
  Jurcevic, P. Demonstration of Quantum Volume 64 on a Superconducting Quantum Computing System. Quantum Sci. Technol. 2021, 6, 025020 DOI: 10.1088/2058-9565/abe519
  There is no corresponding record for this reference.
179. 179
  Pokharel, B.; Lidar, D. A. Demonstration of Algorithmic Quantum Speedup. Phys. Rev. Lett. 2023, 130, 210602, DOI: 10.1103/PhysRevLett.130.210602
  There is no corresponding record for this reference.
180. 180
  Tripathi, V.; Chen, H.; Khezri, M.; Yip, K.-W.; Levenson-Falk, E.; Lidar, D. A. Suppression of Crosstalk in Superconducting Qubits Using Dynamical Decoupling. Phys. Rev. Appl. 2022, 18, 024068 DOI: 10.1103/PhysRevApplied.18.024068
  There is no corresponding record for this reference.
181. 181
  Zhou, Z.; Sitler, R.; Oda, Y.; Schultz, K.; Quiroz, G. Quantum Crosstalk Robust Quantum Control. Phys. Rev. Lett. 2023, 131, 210802, DOI: 10.1103/PhysRevLett.131.210802
  There is no corresponding record for this reference.
182. 182
  Bäumer, E.; Tripathi, V.; Wang, D. S.; Rall, P.; Chen, E. H.; Majumder, S.; Seif, A.; Minev, Z. K. Efficient Long-Range Entanglement using Dynamic Circuits. 2023.
  There is no corresponding record for this reference.
183. 183
  Shirizly, L.; Misguich, G.; Landa, H. Dissipative Dynamics of Graph-State Stabilizers with Superconducting Qubits. Phys. Rev. Lett. 2024, 132, 010601 DOI: 10.1103/PhysRevLett.132.010601
  There is no corresponding record for this reference.
184. 184
  Niu, S.; Todri-Sanial, A.; Bronn, N. T. Multi-qubit dynamical decoupling for enhanced crosstalk suppression. Quantum Science and Technology 2024, 9, 045003 DOI: 10.1088/2058-9565/ad5a37
  There is no corresponding record for this reference.
185. 185
  Evert, B.; Izquierdo, Z. G.; Sud, J.; Hu, H.-Y.; Grabbe, S.; Rieffel, E. G.; Reagor, M. J.; Wang, Z. Syncopated Dynamical Decoupling for Suppressing Crosstalk in Quantum Circuits. 2024; https://arxiv.org/abs/2403.07836 (accessed: 2025-02-23).
  There is no corresponding record for this reference.
186. 186
  Brown, A. F.; Lidar, D. A. Efficient Chromatic-Number-Based Multi-Qubit Decoherence and Crosstalk Suppression. 2024. https://arxiv.org/abs/2406.13901(accessed: 2025-02-23).
  There is no corresponding record for this reference.
187. 187
  Martinis, J. M.; Geller, M. R. Fast adiabatic qubit gates using only σ_z control. Phys. Rev. A 2014, 90, 022307 DOI: 10.1103/PhysRevA.90.022307
  There is no corresponding record for this reference.
188. 188
  Krantz, P.; Kjaergaard, M.; Yan, F.; Orlando, T. P.; Gustavsson, S.; Oliver, W. D. A quantum engineer’s guide to superconducting qubits. Applied Physics Reviews 2019, 6, 021318 DOI: 10.1063/1.5089550
  There is no corresponding record for this reference.
189. 189
  Genov, G. T.; Schraft, D.; Vitanov, N. V.; Halfmann, T. Arbitrarily Accurate Pulse Sequences for Robust Dynamical Decoupling. Phys. Rev. Lett. 2017, 118, 133202, DOI: 10.1103/PhysRevLett.118.133202
  There is no corresponding record for this reference.
190. 190
  Motzoi, F.; Gambetta, J. M.; Rebentrost, P.; Wilhelm, F. K. Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits. Phys. Rev. Lett. 2009, 103, 110501, DOI: 10.1103/PhysRevLett.103.110501
  There is no corresponding record for this reference.
191. 191
  Chow, J. M.; Gambetta, J. M.; Tornberg, L.; Koch, J.; Bishop, L. S.; Houck, A. A.; Johnson, B. R.; Frunzio, L.; Girvin, S. M.; Schoelkopf, R. J. Randomized Benchmarking and Process Tomography for Gate Errors in a Solid-State Qubit. Phys. Rev. Lett. 2009, 102, 090502 DOI: 10.1103/PhysRevLett.102.090502
  There is no corresponding record for this reference.
192. 192
  Albash, T.; Boixo, S.; Lidar, D. A.; Zanardi, P. Quantum adiabatic Markovian master equations. New J. of Phys. 2012, 14, 123016, DOI: 10.1088/1367-2630/14/12/123016
  There is no corresponding record for this reference.
193. 193
  Dann, R.; Levy, A.; Kosloff, R. Time-dependent Markovian quantum master equation. Phys. Rev. A 2018, 98, 052129 DOI: 10.1103/PhysRevA.98.052129
  There is no corresponding record for this reference.
194. 194
  Campos Venuti, L.; Lidar, D. A. Error reduction in quantum annealing using boundary cancellation: Only the end matters. Phys. Rev. A 2018, 98, 022315 DOI: 10.1103/PhysRevA.98.022315
  There is no corresponding record for this reference.
195. 195
  Mozgunov, E.; Lidar, D. Completely positive master equation for arbitrary driving and small level spacing. Quantum 2020, 4, 227, DOI: 10.22331/q-2020-02-06-227
  There is no corresponding record for this reference.
196. 196
  Nathan, F.; Rudner, M. S. Universal Lindblad equation for open quantum systems. Phys. Rev. B 2020, 102, 115109, DOI: 10.1103/PhysRevB.102.115109
  There is no corresponding record for this reference.
197. 197
  Davidović, D. Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation. Quantum 2020, 4, 326, DOI: 10.22331/q-2020-09-21-326
  There is no corresponding record for this reference.
198. 198
  Davidović, D. Geometric-arithmetic master equation in large and fast open quantum systems. Journal of Physics A: Mathematical and Theoretical 2022, 55, 455301, DOI: 10.1088/1751-8121/ac9f30
  There is no corresponding record for this reference.
199. 199
  Kasatkin, V.; Gu, L.; Lidar, D. A. Which differential equations correspond to the Lindblad equation?. Physical Review Research 2023, 5, 043163 DOI: 10.1103/PhysRevResearch.5.043163
  There is no corresponding record for this reference.
200. 200
  Perko, L. Differential Equations and Dynamical Systems; Springer New York: New York, NY, 2001; pp 1– 63.
  There is no corresponding record for this reference.
201. 201
  Geerlings, K.; Leghtas, Z.; Pop, I. M.; Shankar, S.; Frunzio, L.; Schoelkopf, R. J.; Mirrahimi, M.; Devoret, M. H. Demonstrating a Driven Reset Protocol for a Superconducting Qubit. Phys. Rev. Lett. 2013, 110, 120501, DOI: 10.1103/PhysRevLett.110.120501
  There is no corresponding record for this reference.
202. 202
  Goss, N.; Morvan, A.; Marinelli, B.; Mitchell, B. K.; Nguyen, L. B.; Naik, R. K.; Chen, L.; Jünger, C.; Kreikebaum, J. M.; Santiago, D. I.; Wallman, J. J.; Siddiqi, I. High-fidelity qutrit entangling gates for superconducting circuits. Nat. Commun. 2022, 13, 7481, DOI: 10.1038/s41467-022-34851-z
  There is no corresponding record for this reference.
203. 203
  Tripathi, V.; Goss, N.; Vezvaee, A.; Nguyen, L. B.; Siddiqi, I.; Lidar, D. A. Qudit Dynamical Decoupling on a Superconducting Quantum Processor. Phys. Rev. Lett. 2025, 134, 050601 DOI: 10.1103/PhysRevLett.134.050601
  There is no corresponding record for this reference.
204. 204
  Krishnan Vijayan, M.; Paler, A.; Gavriel, J.; Myers, C. R.; Rohde, P. P.; Devitt, S. J. Compilation of algorithm-specific graph states for quantum circuits. Quantum Science and Technology 2024, 9, 025005 DOI: 10.1088/2058-9565/ad1f39
  There is no corresponding record for this reference.
205. 205
  Beverland, M. E.; Murali, P.; Troyer, M.; Svore, K. M.; Hoefler, T.; Kliuchnikov, V.; Low, G. H.; Soeken, M.; Sundaram, A.; Vaschillo, A. Assessing requirements to scale to practical quantum advantage. 2022. https://arxiv.org/abs/2211.07629 (accessed 2025-02-23).
  There is no corresponding record for this reference.
206. 206
  Agrawal, A. A.; Job, J.; Wilson, T. L.; Saadatmand, S. N.; Hodson, M. J.; Mutus, J. Y.; Caesura, A.; Johnson, P. D.; Elenewski, J. E.; Morrell, K. J.; Kemper, A. F. Quantifying fault tolerant simulation of strongly correlated systems using the Fermi-Hubbard model. 2024; https://arxiv.org/abs/2406.06511(accessed: 2025-02-23).
  There is no corresponding record for this reference.
207. 207
  Barends, R. Coherent Josephson Qubit Suitable for Scalable Quantum Integrated Circuits. Phys. Rev. Lett. 2013, 111, 080502 DOI: 10.1103/PhysRevLett.111.080502
  There is no corresponding record for this reference.
208. 208
  Shanto, S.; Kuo, A.; Miyamoto, C.; Zhang, H.; Maurya, V.; Vlachos, E.; Hecht, M.; Shum, C. W.; Levenson-Falk, E. SQuADDS: A Validated Design Database and Simulation Workflow for Superconducting Qubit Design. Quantum 2024, 8, 1465, DOI: 10.22331/q-2024-09-09-1465
  There is no corresponding record for this reference.
209. 209
  Shah, P. A. . qiskit-community/qiskit-metal: Qiskit Metal 0.1.5a1 (2023). 2023; DOI: 10.5281/zenodo.8061781 (accessed 2025-02-23).
  There is no corresponding record for this reference.
210. 210
  Minev, Z. K.; Leghtas, Z.; Reinhold, P.; Mundhada, S. O.; Diringer, A.; Hillel, D. C.; Wang, D. Z.-R.; Facchini, M.; Shah, P. A.; Devoret, M. pyEPR: The energy-participation-ratio (EPR) open- source framework for quantum device design. 2021; DOI: 10.5281/zenodo.4744448 , https://github.com/zlatko-minev/pyEPR; https://pyepr-docs.readthedocs.io/en/latest/ (accessed 2025-02-23).
  There is no corresponding record for this reference.
211. 211
  Chen, Z. Measuring and Suppressing Quantum State Leakage in a Superconducting Qubit. Phys. Rev. Lett. 2016, 116, 020501 DOI: 10.1103/PhysRevLett.116.020501
  There is no corresponding record for this reference.

Benchmarking Quantum Gates and CircuitsClick to copy article linkArticle link copied!

Chemical Reviews

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Abstract

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Special Issue

1. Introduction

2. Gate Errors

2.1. Quantum Dynamical Maps

2.2. Coherent and Incoherent Errors

2.3. Single-Qubit Gate Errors

2.4. Two-Qubit Gate Errors

3. Gate and Circuit Benchmarking Protocols

3.1. Randomized Benchmarking

3.1.1. Standard RB

3.1.2. RB Theory

3.1.3. Interleaved RB

3.2. Quantum Process Tomography

3.2.1. Standard Quantum Process Tomography

Figure 1

3.2.2. Ancilla-Assisted Process Tomography

3.2.3. Direct Characterization of Quantum Dynamics

3.3. Gate Set Tomography

Figure 2

3.4. Direct Fidelity Estimation

Figure 3

3.5. Process Fidelity Estimation

3.6. Cross-Entropy Benchmarking

Figure 4

4. Deterministic Benchmarking

Figure 5

4.1. Qubit Model

4.2. Four Key Parameters

4.3. Protocol

4.4. Experimental Results

Figure 6

Figure 7

4.5. Fidelity with Coherent Errors

4.6. Open System Model

4.7. Sensitivity to Coherent Errors

Figure 8

4.8. Interplay of T1 Asymmetry and Gates

Figure 9

4.9. Derivations: Fidelities of the YY and XX Sequences

4.9.1. Background

4.9.2. YY Sequence

Exact Solution for Δerr = 0

Perturbation Theory for Δerr = 0

Perturbation Theory for Δerr ≠ 0

4.9.3. XX̅ Sequence

4.10. Finite Temperature

4.10.1. Numerical

4.10.2. Experimental

4.10.3. Analytical

4.11. Leakage

Figure 10

4.12. DB Summary and Future Research Directions

5. Conclusions and Outlook

Author Information

Biographies

Vinay Tripathi

Daria Kowsari

Kumar Saurav

Haimeng Zhang

Eli M. Levenson-Falk

Daniel A. Lidar

Acknowledgments

Appendix

A. Device Design and Experimental Setup

B. Conventional Single-Qubit Gate Calibration

C. ζ’s Components

References

Cited By

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Benchmarking Quantum Gates and Circuits
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4.8. Interplay of T₁ Asymmetry and Gates

Exact Solution for Δ_err = 0

Perturbation Theory for Δ_err = 0

Perturbation Theory for Δ_err ≠ 0