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Quantum Electronic Structure

Reference-State Error Mitigation: A Strategy for High Accuracy Quantum Computation of Chemistry
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  • Phalgun Lolur
    Phalgun Lolur
    Department of Chemistry and Chemical Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
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    Orcidhttps://orcid.org/0000-0002-3409-3415
  • Mårten Skogh
    Mårten Skogh
    Department of Chemistry and Chemical Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
    Data Science & Modelling, Pharmaceutical Science, R&D, AstraZeneca, SE-431 83 Mölndal, Gothenburg, Sweden
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  • Werner Dobrautz
    Werner Dobrautz
    Department of Chemistry and Chemical Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
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    Orcidhttps://orcid.org/0000-0001-6479-1874
  • Christopher Warren
    Christopher Warren
    Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
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  • Janka Biznárová
    Janka Biznárová
    Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
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  • Amr Osman
    Amr Osman
    Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
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  • Giovanna Tancredi
    Giovanna Tancredi
    Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
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  • Göran Wendin
    Göran Wendin
    Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
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  • Jonas Bylander
    Jonas Bylander
    Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
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  • Martin Rahm*
    Martin Rahm
    Department of Chemistry and Chemical Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
    *Email: [email protected]
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    Orcidhttps://orcid.org/0000-0001-7645-5923
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Journal of Chemical Theory and Computation

Cite this: J. Chem. Theory Comput. 2023, 19, 3, 783–789
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https://doi.org/10.1021/acs.jctc.2c00807
Published January 27, 2023

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Abstract

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Decoherence and gate errors severely limit the capabilities of state-of-the-art quantum computers. This work introduces a strategy for reference-state error mitigation (REM) of quantum chemistry that can be straightforwardly implemented on current and near-term devices. REM can be applied alongside existing mitigation procedures, while requiring minimal postprocessing and only one or no additional measurements. The approach is agnostic to the underlying quantum mechanical ansatz and is designed for the variational quantum eigensolver. Up to two orders-of-magnitude improvement in the computational accuracy of ground state energies of small molecules (H2, HeH+, and LiH) is demonstrated on superconducting quantum hardware. Simulations of noisy circuits with a depth exceeding 1000 two-qubit gates are used to demonstrate the scalability of the method.

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Introduction

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Quantum computers hold a potential for solving problems that are intractable on current and future computers. (1,2) Quantum chemistry is one of the research areas where quantum advantage is expected in the near future. (3−6) One of the major challenges in realizing practical quantum computation of chemistry is the sensitivity of quantum devices to noise. Errors due to noise can be caused by several factors such as spontaneous emission, control and measurement imperfection, and unwanted coupling with the environment. (7) Whereas reliable error correction is expected in future quantum computers, such fault-tolerant machines will put high demands on both quality and number of physical qubits. (4) Increasingly robust hybrid algorithms (2,8−10) are being designed for quantum chemistry on near-term, noisy intermediate-scale quantum (NISQ) devices. (11) Unfortunately, noise causes such algorithms to produce results, such as energies of molecules, that are of relatively low quality, even as they rely on shallow quantum circuits. (12−15) We will return to discuss how one can define quality in terms of accuracy and precision in this context.
The general challenge of noise in quantum hardware has motivated the development of several methods for error mitigation: (16) readout/measurement error mitigation, (17) zero noise/Richardson extrapolation, (18,19) Clifford data regression, (20) training by fermionic linear optics (TFLO), (21) rescaling as per Arute et al., (22) probabilistic error cancellation, (23) quantum subspace expansion, (24) postselection, (25) McWeeny purification, (26) virtual state distillation, (27) and symmetry verification (28) are some examples of techniques exploited to improve the quality of measurements of encoded Hamiltonians through pre- or postprocessing. Some of these techniques have been shown to offer improvements when computing energies of small molecules with variational algorithms. (26,29) A combination of mitigation strategies is often a good approach to minimize errors.
In this study, we report on a chemistry-inspired error-mitigation strategy that can be combined with any variant of the variational quantum eigensolver (9,30) (VQE). Our approach, reference-state error mitigation (REM), relies on postprocessing that can be readily performed on a classical computer. The method is applicable across a wide range of noise intensities and is low-cost in that it requires an overhead of at most one additional VQE energy evaluation. REM can readily be employed together with other error mitigation methods, and throughout this work we additionally use readout mitigation, which corrects for hardware-specific nonideal correlation between prepared and measured states. (17,22) Readout mitigation works by performing an initial calibration against a subset of Clifford gates, whereby known states are prepared and measured; see the Supporting Information (SI).
The evaluation of computational accuracy in this work should not be confused with chemical accuracy. (31) We here use the term computational accuracy specifically when comparing results of a quantum calculation with the exact solution at that same level of theory. Computational accuracy then, in the context of VQE calculations, refers to how accurate a given VQE problem is solved with respect to the given Hamiltonian and ansatz. This accuracy can only be quantified so long as it is possible to solve the problem without noise, e.g., using conventional quantum chemistry (which we can still do; discussing the limits of conventional quantum chemistry methods is outside the scope of this work). Practical implementations of VQE on real hardware currently suffer from drastic deficiencies in level of theory, basis set size, (32) proper consideration of the physical environment (e.g., solvent effects), and dynamical effects. These limitations keep quantum computation (including our own) from accurately predicting real chemical processes. On the other hand, chemical accuracy is the correct term for what is required to make realistic predictions and is commonly defined as an error of 1 kcal/mol (∼1.6 millihartree) from the exact solution. (9,15,26,33) We encourage the community to use the appropriate terminology. The hunt for chemical accuracy in the NISQ era is far from over.

Methods

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The goal of the VQE algorithm is to minimize the electronic energy with respect to a set of quantum circuit parameters, i.e.,
EVQE=minθ→⁡E(θ→)=minθ→⟨Ψ(θ→)|H^|Ψ(θ→)⟩
(1)
where θ⃗ = [θ1, θ2, ..., θn], Ĥ is the molecular Hamiltonian, and |Ψ(θ⃗)⟩ represents the parametrized trial state generated by the VQE circuit. The VQE energy, EVQE(θ⃗), can therefore be thought of as living on an n-dimensional surface in parameter space. A one-dimensional representation of such a surface is shown in orange at the top of Figure 1. The EVQE(θ⃗) surface is associated with some degree of systematic and random noise that can only be partially removed by accounting for state-preparation and measurement errors through readout mitigation.

Figure 1

Figure 1. (a) One-dimensional representation of the electronic energy E as a function of quantum circuit parameters θ⃗. The REM approach can be explained as a four-step process: (1) A computationally tractable reference solution (such as Hartree–Fock) is computed on a classical computer. (2) A quantum measurement of the VQE surface (orange) is made using parameters corresponding to the reference solution. (3) The difference in energy calculated for the reference solution on classical and quantum hardware defines an error, ΔEREM. (4) The error estimate is assumed to be systematic and is used to correct the VQE surface in the proximity of the reference coordinate. The resulting REM corrected surface and the exact (noise-free) solution are represented by solid green and dashed red lines, respectively. Blue dots represent the coordinates of the reference calculations, while diamonds indicate minima of the different energy landscapes. (b) The inset shows the remaining error after application of REM. A possible difference in the location of the minima is indicated by θ⃗-shift (b). The difference in energy between the VQE and REM minima is indicated by ΔEerror,REM.

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As the name suggests, the REM method rests on an appropriate choice of a reference wave function, or reference state. We recommend the reference state is (a) chemically motivated, i.e., likely physically similar to the sought state, and (b) fast (or at least viable) to evaluate using a classical computer. These properties are often also desired in the choice of initial guess for VQE, thus the initial point for optimization is commonly a good choice of reference state. The Hartree–Fock state is a practical example of an often-suitable reference wave function that is based on a computationally efficient mean-field description of the electronic potential. We rely on Hartree–Fock as the reference state in this work, but we will investigate different reference choices, better suited to more strongly correlated problems, in future work.
For the REM method to be practically useful, the cost associated with calculating the reference state has to be lower than the actual VQE calculation. One way to evaluate the relative cost of classical as compared to quantum computation is to compare their respective computational complexity. A lower bound estimate of the cost of VQE is the number of measurements required to evaluate the energy. In an ideal scenario, i.e., without noise, such measurements approximately scale as O(n4), where n is the number of basis functions. In contrast, the cost of conventional Hartree–Fock calculations have a practical scaling between O(n2) and O(n4). (34)
Once the parametrized reference state |Ψ(θ⃗ref)⟩ is prepared, a determination of the resulting energy error ΔEREM at the reference parameters can be made,
ΔEREM=EVQE(θ→ref)−Eexact(θ→ref)
(2)
where Eexact(θ⃗ref) is the exact solution (up to numerical precision) for the reference state, evaluated on a classical computer. EVQE(θ⃗ref) refers to the energy evaluated from measurements on a quantum computer at the reference parameter value, θ⃗ref. The idea of approaching error mitigation by comparing noisy measurements to noise-free tractable classical calculations shares some commonality with mitigation techniques such as Clifford data regression, TFLO, and the rescaling technique of Arute et al. (22) However, our method is distinct from these methods in that REM only relies on the use of a single conventional calculation to generate a reference state. Because REM incorporates mitigation through the choice of a chemically and physically motivated initial guess for the VQE algorithm it does not require training on a large number of measured expectation values, as Clifford data regression and TFLO do. REM also considers the effects of both circuit depth and composition, which are often overlooked by other methods.
It is important to make the distinction between the initial state in the VQE calculation and the reference state, which need not be the same. The reference state can either be a part of the VQE optimization or be prepared and measured separately from the variational procedure. Provided that the reference state is also used as an initial guess for the VQE algorithm, it is possible to perform REM without incurring any additional measurement cost.
The exact energy at any arbitrary coordinate, Eexact(θ⃗), can be expressed as
Eexact(θ→)=EVQE(θ→)−ΔEREM−ΔEp(θ→)
(3)
where ΔEp(θ⃗) includes any parameter-dependence of noise present and ΔEp(θ⃗ref) = 0. The underlying assumption of the REM method is that such parameter dependence of the noise is negligible close to the reference geometry, i.e., limΔθ→→0⁡ΔEp(θ→)=0 where Δθ⃗ = | θ⃗ – θ⃗ref| . In other words, the effectiveness of the REM approach can be assumed dependent on the Euclidean distance of the reference state to the exact solution |θ⃗exact – θ⃗ref|, given that both are in the same convex region of the energy surface. When this approximation fails, noise can shift features in the energy surface, such as the optimal coordinates identified using the VQE algorithm, θ⃗min,VQE, away from the true minimum, θ⃗min,exact(Figure 1b), resulting in a θ⃗-shift. When evaluating our method, we will not quantify ΔEp(θ⃗) but instead compare energies obtained for the two minima on the exact and the VQE surface,
ΔEerror,VQE=EVQE(θ→min,VQE)−Eexact(θ→min,exact)
(4)
In eq 4, Eexact(θ⃗min,exact) is the exact solution obtained by an ideal noise-free VQE optimization and EVQE(θ⃗min,VQE) is the energy of a converged noisy VQE optimization. The error remaining after applying REM to a converged noisy VQE optimization is
ΔEerror,REM=ΔEREM−ΔEerror,VQE
(5)

Results and Discussion

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To assess the reliability of REM, we have implemented it for the ground state energy computation of small molecules on two current NISQ devices, the ibmq_quito of IBMQ and the Särimner device of Chalmers University. Details of hardware, circuits, measurements and estimates on confidence bounds are provided in the SI. Table 1 shows an amalgamation of our measurement results for the ground state energy of the hydrogen molecule (H2), helium hydride (HeH+), and lithium hydride (LiH). The ansatze (3) used for the H2 and HeH+ molecules are chemistry-inspired and based on unitary coupled cluster theory, (35) whereas a hardware-efficient ansatz is used for LiH. Table 1 also includes results of simulations of LiH and beryllium hydride (BeH2). The latter circuits are substantially larger than what is feasible on current devices as they would incur insurmountable errors due to noise. Combined, this test set ranges from a two-qubit circuit with just 1 two-qubit gate for H2, to a six-qubit circuit with 1096 two-qubit gates for BeH2 (Table 2). Our simulations of BeH2 contains 26 variational parameters (Table 2).
Table 1. Total Ground State Energies of Molecules at Experimental Equilibrium Distancesa, without and with the Application of REMb
MoleculeaEexact(θ⃗min)EVQE(θ⃗min,VQE)EREMΔEerror,VQEΔEerror,REM
H2c–1.1373–1.1085(60)–1.13550.029(6)0.002(8)
HeH+d–2.8542–2.825(4)–2.853(6)0.029(4)0.003(6)
LiHd–7.8787–7.599(33)–7.852(62)0.280(33)0.029(62)
LiHe–7.8811–7.360(4)–7.871(7)0.521(4)0.011(7)
BeH2e–15.5895–13.987(5)–15.563(10)1.602(5)0.0263(10)
a

Calculations refer to experimental bond distances from the National Institute of Standards and Technology (NIST). (36) Details on measurement and confidence bounds are provided in the SI.

b

Readout mitigation has been applied for all VQE calculations. All energies are given in hartree. Associated standard deviations are provided for all errors.

c

Run on Chalmers Särimner with 5000 samples as a single point measurement without optimization. The sample size of 5000 is motivated by our previous experience with the device. The computation on the Särimner device was performed as a complete sweep of the single variational parameter in the circuit, and not as a VQE optimization. Therefore, the energy variance must be approximated through other means, which we describe in the SI.

d

Run on ibmq_quito with 8192 samples, the maximum allowed.

e

Simulated results using a noise model from ibmq_athens.

Table 1 shows how the application of REM reduces the error by up to 2 orders of magnitude compared to regular VQE when used together with readout mitigation. Without readout mitigation, VQE errors are substantially larger (Tables S3 and S6). The examples in Table 1 are sorted by increasing circuit depth (see also Table 2 for details), which indicate both the robustness and scalability of the approach. The remaining error after mitigation is consistently on the order of millihartree, and the magnitude of the REM correction grows with the complexity of the quantum circuit (cf. H2 vs BeH2 in Table 1).
In principle, unsuitable choices of reference states combined with significant parameter dependence of noise, ΔEp(θ⃗) ≉ 0, might result in overcorrection of the measured VQE energy, taking the energy below the true minimum, as can be seen in some results for the dissociation curve of HeH+ (Figures 2 and 3). Nevertheless, we note that REM consistently improves the measured energies, even at relatively high noise levels (Figure 3) and drastically improves the computational accuracy for all calculations summarized in Table 1.

Figure 2

Figure 2. Top: Potential energy surfaces for the dissociation of H2 and HeH+. Exact noise-free solutions from state-vector simulations are represented by black lines. Regular VQE energies obtained using a quantum computer are shown as blue dots. Results following readout mitigation and REM are shown as orange crosses and green squares, respectively. The combination of both readout mitigation and REM is shown as red stars. Measurements for H2 and HeH+ were performed on Chalmers Särimner and ibmq_quito, respectively. Bottom: error of the different approaches relative to the exact solution in the given minimal basis set. The gray region corresponds to an error of 1.6 millihartree (1 kcal/mol) with respect to the corresponding noise-free calculations.

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Figure 3

Figure 3. Absolute errors as a function of increasing depolarizing errors for ground state calculations of H2 and HeH+. The two-qubit depolarizing error of the Chalmers Särimner device is indicated by a vertical line for reference. Bottom inset: Energy errors after application of REM are largely independent of the noise level, and the computational accuracy is consistently close to or below 1.6 millihartree (1 kcal/mol) for these molecules.

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Testing the Limits of REM

Table 1 demonstrates the effectiveness of REM when applied to molecules in their equilibrium geometry. The test set is small, in practice limited to what is feasible to run on current NISQ hardware. These geometries also represent situations where the degree of electron correlation is relatively low.
To investigate how the REM method performs out of equilibrium, we show in Figure 2 the bond dissociation of H2 and HeH+. For HeH+, which dissociates to He and an isolated proton─a state well described by a single-reference Hartree–Fock description─the REM method provides highly accurate results across the entire binding curve. For H2 on the other hand, the effectiveness of the current implementation of REM decreases in regions where the Hartree–Fock state offers a poor description, such as the stretched H2 bond. Nevertheless, the method consistently provides a substantial improvement across the potential energy surface. A more suitable description of the partially broken bond of H2 should ideally account for the static correlation arising due to near degeneracy of multiple states; i.e., it would require a multireference (MR) or open-shell (OS) reference. We will investigate the use of such MR/OS states and an adaptive choice of the most suitable single reference (SR) state within the REM framework in an upcoming study.
Figure 2 also illustrates the effect of readout mitigation, (17) which we use per default in all measurements and that we recommend together with REM. Other mitigation strategies may, in principle, also be combined with REM.
The robustness of REM was further evaluated by performing simulations of H2 and HeH+ while varying the noise level. Noise was introduced in these simulations by modeling imperfect gate-fidelities as single (S) and two-qubit (T) depolarizing errors, connected through a linear relationship, S = 0.1 T (see the SI). This kind of noise modeling enables straightforward comparison with error rates on physical quantum devices (Figure 3). REM is shown to be effective despite the steady increase in single- and two-qubit depolarizing error rates.

Conclusions

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In this work we demonstrate an error mitigation strategy applicable to quantum chemical computations on NISQ devices. The REM method relies on accurately determining the error in energy due to hardware and environmental noise for a reference wave function that can be feasibly evaluated on a classical computer. The underlying assumption of REM is a negligible dependence of noise on circuit parameters in the vicinity of this reference wave function. In this work Hartree–Fock references are used, which describes the physics of the molecular states well enough for REM to perform effectively. The REM method is shown to drastically improve the computational accuracy at which total energies of molecules can be computed using current quantum hardware. REM is well suited for calculations with significant amounts of noise and improve calculated energies in all herein tested cases.
The performance of REM is dependent on the quality of the supplied reference state. A Hartree–Fock (mean-field) solution is expected to provide a sufficient reference for molecules that do not exhibit large multireference character. We will investigate the use of references based on multireference and open-shell states, better suited for more strongly correlated problems, in an upcoming study.
In our herein studied problems, the computational accuracy is improved by up to 2 orders of magnitude after application of REM. However, in the presence of substantial quantum circuit parameter dependence of noise it cannot be ruled out that REM may underestimate the true energy. The nonvariational nature of error mitigation strategies remains a problem to be solved. No single mitigation technique will completely resolve the issue of noise, and REM is no exception. One strength of REM is its ability to be combined with other error mitigation techniques without incurring additional cost. REM does not incur meaningful additional classical or quantum computational overhead and can be used to reduce errors on near-term devices by orders of magnitude when running VQE calculations. Because error rates vary both between NISQ devices and between circuits, it is not currently productive to rely on error cancellation, i.e., systematic errors inherent in quantum chemical levels of theory, when evaluating relative energies of chemical transformations. By enabling more precise evaluations of molecular total energies, REM moves us toward meaningful relative comparisons and toward chemical accuracy.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.2c00807.

  • Details of the Chalmers device; description of read out error mitigation; details for H2, HeH+, LiH, and BeH2 calculations, including quantum circuits and Hamiltonians; simulation details for LiH and BeH2; calibration details for ibmq_quito device; and description of depolarizing noise model used for simulations (PDF)

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Author Information

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  • Corresponding Author
  • Authors
    • Phalgun Lolur - Department of Chemistry and Chemical Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden;  Present Address: (P.L.) National Quantum Computing Centre (NQCC), Science and Technology Facilities Council (STFC), UK Research and Innovation (UKRI), Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire OX11 0QX, United Kingdom;  Orcidhttps://orcid.org/0000-0002-3409-3415
    • Mårten Skogh - Department of Chemistry and Chemical Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden;  Data Science & Modelling, Pharmaceutical Science, R&D, AstraZeneca, SE-431 83 Mölndal, Gothenburg, Sweden
    • Werner Dobrautz - Department of Chemistry and Chemical Engineering, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden;  Orcidhttps://orcid.org/0000-0001-6479-1874
    • Christopher Warren - Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
    • Janka Biznárová - Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
    • Amr Osman - Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
    • Giovanna Tancredi - Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
    • Göran Wendin - Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
    • Jonas Bylander - Department of Microtechnology and Nanoscience MC2, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
  • Author Contributions

    Conceptualization: P.L., M.S., M.R. Funding acquisition: G.W., J.By. Investigation (Theoretical): P.L., M.S. Investigation (Experimental): C.W., G.T. Device Fabrication: J.Bi., A.O. Supervision: M.R., J.By., G.W. Writing: M.R., P.L., M.S., W.D., C.W., J.By.

    Author Contributions

    P.L. and M.S. contributed equally to this work.

  • Funding

    This research has been supported by funding from the Wallenberg Center for Quantum Technology (WACQT) and from the EU Flagship on Quantum Technology H2020-FETFLAG-2018-03 Project 820363 OpenSuperQ. W.D. acknowledges funding from the European Union’s Horizon Europe research and innovation program under the Marie Skłodowska-Curie grant agreement No. 101062864. This research relied on computational resources provided by the Swedish National Infrastructure for Computing (SNIC) at C3SE, NSC, and PDC partially funded by the Swedish research council through Grant Agreement No. 2018-05973.

  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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We also acknowledge experimental assistance from Christian Križan and Andreas Bengtsson as well as insights from Jorge Fernández Pendás at Chalmers University of Technology. The Chalmers device was made at Myfab Chalmers; it was packaged in a holder and printed circuit board with original designs shared by the Quantum Device Lab at ETH Zürich.

Appendix: Circuit Complexity Details

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Table 2 is presented here.
Table 2. Comparison of Circuit Complexities for H2, HeH+, LiH, and BeH2 Listed in Order of Increasing Complexitya
MoleculeAnsatzQubitsDepthTwo-Qubit GatesParameters
H2Simplified UCCD2511
HeH+UCCSD21442
LiHHardware-efficient4968
LiHUCCSD42751728
BeH2UCCSD61480109626
a

Details are provided in the SI.

Abbreviations

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BeH2

beryllium hydride

H2

hydrogen molecule

HeH+

helium hydride cation

LiH

lithium hydride

NISQ

noisy intermediate-scale quantum

NIST

National Institute for Standards and Technology

REM

reference-state error mitigation

TFLO

training by Fermionic linear optics

VQE

variational quantum eigensolver

MR

multireference

OS

open-shell

SR

single-reference.

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    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 (19), 10856– 10915,  DOI: 10.1021/acs.chemrev.8b00803
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    Quantum Chemistry in the Age of Quantum Computing
    Cao, Yudong; Romero, Jonathan; Olson, Jonathan P.; Degroote, Matthias; Johnson, Peter D.; Kieferova, Maria; Kivlichan, Ian D.; Menke, Tim; Peropadre, Borja; Sawaya, Nicolas P. D.; Sim, Sukin; Veis, Libor; Aspuru-Guzik, Alan
    Chemical Reviews (Washington, DC, United States) (2019), 119 (19), 10856-10915CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
    A review. Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chem. communities over the past century. Although many approxn. methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging complexity landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chem. such as the electronic structure of mols. In the past two decades significant advances have been made in developing algorithms and phys. hardware for quantum computing, heralding a revolution in simulation of quantum systems. This article is an overview of the algorithms and results that are relevant for quantum chem. The intended audience is both quantum chemists who seek to learn more about quantum computing, and quantum computing researchers who would like to explore applications in quantum chem.
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  5. 5
    Lanyon, B. P.; Whitfield, J. D.; Gillett, G. G.; Goggin, M. E.; Almeida, M. P.; Kassal, I.; Biamonte, J. D.; Mohseni, M.; Powell, B. J.; Barbieri, M.; Aspuru-Guzik, A.; White, A. G. Towards Quantum Chemistry on a Quantum Computer. Nat. Chem. 2010, 2 (2), 106– 111,  DOI: 10.1038/nchem.483
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    Towards quantum chemistry on a quantum computer
    Lanyon, B. P.; Whitfield, J. D.; Gillett, G. G.; Goggin, M. E.; Almeida, M. P.; Kassal, I.; Biamonte, J. D.; Mohseni, M.; Powell, B. J.; Barbieri, M.; Aspuru-Guzik, A.; White, A. G.
    Nature Chemistry (2010), 2 (2), 106-111CODEN: NCAHBB; ISSN:1755-4330. (Nature Publishing Group)
    Exact first-principles calcns. of mol. properties are currently intractable because their computational cost grows exponentially with both the no. of atoms and basis set size. A soln. is to move to a radically different model of computing by building a quantum computer, which is a device that uses quantum systems themselves to store and process data. Here we report the application of the latest photonic quantum computer technol. to calc. properties of the smallest mol. system: the mol. in a minimal basis. We calc. the complete energy spectrum to 20 bits of precision and discuss how the technique can be expanded to solve large-scale chem. problems that lie beyond the reach of modern supercomputers. These results represent an early practical step toward a powerful tool with a broad range of quantum-chem. applications.
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  6. 6
    Cerezo, M.; Arrasmith, A.; Babbush, R.; Benjamin, S. C.; Endo, S.; Fujii, K.; McClean, J. R.; Mitarai, K.; Yuan, X.; Cincio, L.; Coles, P. J. Variational Quantum Algorithms. Nature Reviews Physics 2021, 3 (9), 625– 644,  DOI: 10.1038/s42254-021-00348-9
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    Unruh, W. G. Maintaining Coherence in Quantum Computers. Phys. Rev. A (Coll Park) 1995, 51 (2), 992– 997,  DOI: 10.1103/PhysRevA.51.992
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    Hadfield, S.; Wang, Z.; O’Gorman, B.; Rieffel, E. G.; Venturelli, D.; Biswas, R. From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz. Algorithms 2019, 12 (2), 34,  DOI: 10.3390/a12020034
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    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 Quantum Processor. Nat. Commun. 2014, 5 (1), 4213,  DOI: 10.1038/ncomms5213
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    A variational eigenvalue solver on a photonic quantum processor
    Peruzzo, Alberto; McClean, Jarrod; Shadbolt, Peter; Yung, Man-Hong; Zhou, Xiao-Qi; Love, Peter J.; Aspuru-Guzik, Alan; O'Brien, Jeremy L.
    Nature Communications (2014), 5 (), 4213CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
    Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the phys. dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estn. algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state prepn. based on ansatze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We exptl. demonstrate the feasibility of this approach with an example from quantum chem.-calcg. the ground-state mol. energy for He-H+. The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future.
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  10. 10
    Farhi, E.; Goldstone, J.; Gutmann, S. A Quantum Approximate Optimization Algorithm. arXiv , 1411.4028 [quant-ph], November 14, 2014, https://arxiv.org/abs/1411.4028 (accessed 2023-01-10).
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    Preskill, J. Quantum Computing in the NISQ Era and Beyond. Quantum 2018, 2 (July), 79,  DOI: 10.22331/q-2018-08-06-79
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    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 (3), 031022,  DOI: 10.1103/PhysRevX.8.031022
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    Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator
    Hempel, Cornelius; Maier, Christine; Romero, Jonathan; McClean, Jarrod; Monz, Thomas; Shen, Heng; Jurcevic, Petar; Lanyon, Ben P.; Love, Peter; Babbush, Ryan; Aspuru-Guzik, Alan; Blatt, Rainer; Roos, Christian F.
    Physical Review X (2018), 8 (3), 031022CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
    Quantum-classical hybrid algorithms are emerging as promising candidates for near-term practical applications of quantum information processors in a wide variety of fields ranging from chem. to physics and materials science. We report on the exptl. implementation of such an algorithm to solve a quantum chem. problem, using a digital quantum simulator based on trapped ions. Specifically, we implement the variational quantum eigensolver algorithm to calc. the mol. ground-state energies of two simple mols. and exptl. demonstrate and compare different encoding methods using up to four qubits. Furthermore, we discuss the impact of measurement noise as well as mitigation strategies and indicate the potential for adaptive implementations focused on reaching chem. accuracy, which may serve as a cross-platform benchmark for multiqubit quantum simulators.
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    Sawaya, N. P. D.; Smelyanskiy, M.; McClean, J. R.; Aspuru-Guzik, A. Error Sensitivity to Environmental Noise in Quantum Circuits for Chemical State Preparation. J. Chem. Theory Comput 2016, 12 (7), 3097– 3108,  DOI: 10.1021/acs.jctc.6b00220
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    13
    Error Sensitivity to Environmental Noise in Quantum Circuits for Chemical State Preparation
    Sawaya, Nicolas P. D.; Smelyanskiy, Mikhail; McClean, Jarrod R.; Aspuru-Guzik, Alan
    Journal of Chemical Theory and Computation (2016), 12 (7), 3097-3108CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Calcg. mol. energies is likely to be one of the first useful applications to achieve quantum supremacy, performing faster on a quantum than a classical computer. However, if future quantum devices are to produce accurate calcns., errors due to environmental noise and algorithmic approxns. need to be characterized and reduced. In this study, we use the high performance qHiPSTER software to investigate the effects of environmental noise on the prepn. of quantum chem. states. We simulated 18 16-qubit quantum circuits under environmental noise, each corresponding to a unitary coupled cluster state prepn. of a different mol. or mol. configuration. Addnl., we analyze the nature of simple gate errors in noise-free circuits of up to 40 qubits. We find that, in most cases, the Jordan-Wigner (JW) encoding produces smaller errors under a noisy environment as compared to the Bravyi-Kitaev (BK) encoding. For the JW encoding, pure dephasing noise is shown to produce substantially smaller errors than pure relaxation noise of the same magnitude. We report error trends in both mol. energy and electron particle no. within a unitary coupled cluster state prepn. scheme, against changes in nuclear charge, bond length, no. of electrons, noise types, and noise magnitude. These trends may prove to be useful in making algorithmic and hardware-related choices for quantum simulation of mol. energies.
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  14. 14
    Piveteau, C.; Sutter, D.; Bravyi, S.; Gambetta, J. M.; Temme, K. Error Mitigation for Universal Gates on Encoded Qubits. Phys. Rev. Lett. 2021, 127 (20), 200505,  DOI: 10.1103/PhysRevLett.127.200505
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    14
    Error Mitigation for Universal Gates on Encoded Qubits
    Piveteau, Christophe; Sutter, David; Bravyi, Sergey; Gambetta, Jay M.; Temme, Kristan
    Physical Review Letters (2021), 127 (20), 200505CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
    The Eastin-Knill theorem states that no quantum error-correcting code can have a universal set of transversal gates. For Calderbank-Shor-Steane codes that can implement Clifford gates transversally, it suffices to provide one addnl. non-Clifford gate, such as the T gate, to achieve universality. Common methods to implement fault-tolerant T gates, e.g., magic state distn., generate a significant hardware overhead that will likely prevent their practical usage in the near-term future. Recently, methods have been developed to mitigate the effect of noise in shallow quantum circuits that are not protected by error correction. Error mitigation methods require no addnl. hardware resources but suffer from a bad asymptotic scaling and apply only to a restricted class of quantum algorithms. In this Letter, we combine both approaches and show how to implement encoded Clifford+T circuits where Clifford gates are protected from noise by error correction while errors introduced by noisy encoded T gates are mitigated using the quasiprobability method. As a result, Clifford+T circuits with a no. of T gates inversely proportional to the phys. noise rate can be implemented on small error-cor. devices without magic state distn. We argue that such circuits can be out of reach for state-of-the-art classical simulation algorithms.
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  15. 15
    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 (7671), 242– 246,  DOI: 10.1038/nature23879
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    Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets
    Kandala, Abhinav; Mezzacapo, Antonio; Temme, Kristan; Takita, Maika; Brink, Markus; Chow, Jerry M.; Gambetta, Jay M.
    Nature (London, United Kingdom) (2017), 549 (7671), 242-246CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
    Quantum computers can be used to address electronic-structure problems and problems in materials science and condensed matter physics that can be formulated as interacting fermionic problems, problems which stretch the limits of existing high-performance computers. Finding exact solns. to such problems numerically has a computational cost that scales exponentially with the size of the system, and Monte Carlo methods are unsuitable owing to the fermionic sign problem. These limitations of classical computational methods have made solving even few-atom electronic-structure problems interesting for implementation using medium-sized quantum computers. Yet exptl. implementations have so far been restricted to mols. involving only hydrogen and helium. Here we demonstrate the exptl. optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms, detg. the ground-state energy for mols. of increasing size, up to BeH2. We achieve this result by using a variational quantum eigenvalue solver (eigensolver) with efficiently prepd. trial states that are tailored specifically to the interactions that are available in our quantum processor, combined with a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine. We demonstrate the flexibility of our approach by applying it to a problem of quantum magnetism, an antiferromagnetic Heisenberg model in an external magnetic field. In all cases, we find agreement between our expts. and numerical simulations using a model of the device with noise. Our results help to elucidate the requirements for scaling the method to larger systems and for bridging the gap between key problems in high-performance computing and their implementation on quantum hardware.
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    Suzuki, Y.; Endo, S.; Fujii, K.; Tokunaga, Y. Quantum Error Mitigation as a Universal Error Reduction Technique: Applications from the NISQ to the Fault-Tolerant Quantum Computing Eras. PRX Quantum 2022, 3 (1), 010345  DOI: 10.1103/PRXQuantum.3.010345
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    Bravyi, S.; Sheldon, S.; Kandala, A.; McKay, D. C.; Gambetta, J. M. Mitigating Measurement Errors in Multiqubit Experiments. Phys. Rev. A (Coll Park) 2021, 103 (4), 042605,  DOI: 10.1103/PhysRevA.103.042605
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    Li, Y.; Benjamin, S. C. Efficient Variational Quantum Simulator Incorporating Active Error Minimization. Phys. Rev. X 2017, 7 (2), 021050,  DOI: 10.1103/PhysRevX.7.021050
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    18
    Efficient variational quantum simulator incorporating active error minimization
    Li, Ying; Benjamin, Simon C.
    Physical Review X (2017), 7 (2), 021050/1-021050/14CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
    One of the key applications for quantum computers will be the simulation of other quantum systems that arise in chem., materials science, etc., in order to accelerate the process of discovery. It is important to ask the following question: Can this simulation be achieved using near-future quantum processors, of modest size and under imperfect control, or must it await the more distant era of large-scale fault-tolerant quantum computing. Here, we propose a variational method involving closely integrated classical and quantum coprocessors. We presume that all operations in the quantum coprocessor are prone to error. The impact of such errors is minimized by boosting them artificially and then extrapolating to the zero-error case. In comparison to a more conventional optimized Trotterization technique, we find that our protocol is efficient and appears to be fundamentally more robust against error accumulation.
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    Giurgica-Tiron, T.; Hindy, Y.; Larose, R.; Mari, A.; Zeng, W. J. Digital Zero Noise Extrapolation for Quantum Error Mitigation. Proceedings - IEEE International Conference on Quantum Computing and Engineering, QCE 2020; IEEE: 2020; pp 306– 316.
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    Czarnik, P.; Arrasmith, A.; Coles, P. J.; Cincio, L. Error Mitigation with Clifford Quantum-Circuit Data. Quantum 2021, 5, 592,  DOI: 10.22331/q-2021-11-26-592
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    Montanaro, A.; Stanisic, S. Error Mitigation by Training with Fermionic Linear Optics. arXiv, 2102.02120 [quant-ph], February 2021, http://arxiv.org/abs/2102.02120 (accessed 2023-01-09).
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    Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J. C.; Barends, R.; Bengtsson, A.; Boixo, S.; Broughton, M.; Buckley, B. B.; Buell, D. A.; Burkett, B.; Bushnell, N.; Chen, Y.; Chen, Z.; Chen, Y.-A.; Chiaro, B.; Collins, R.; Cotton, S. J.; Courtney, W.; Demura, S.; Derk, A.; Dunsworth, A.; Eppens, D.; Eckl, T.; Erickson, C.; Farhi, E.; Fowler, A.; Foxen, B.; Gidney, C.; Giustina, M.; Graff, R.; Gross, J. A.; Habegger, S.; Harrigan, M. P.; Ho, A.; Hong, S.; Huang, T.; Huggins, W.; Ioffe, L. B.; Isakov, S. v; Jeffrey, E.; Jiang, Z.; Jones, C.; Kafri, D.; Kechedzhi, K.; Kelly, J.; Kim, S.; Klimov, P. v; Korotkov, A. N.; Kostritsa, F.; Landhuis, D.; Laptev, P.; Lindmark, M.; Lucero, E.; Marthaler, M.; Martin, O.; Martinis, J. M.; Marusczyk, A.; McArdle, S.; McClean, J. R.; McCourt, T.; McEwen, M.; Megrant, A.; Mejuto-Zaera, C.; Mi, X.; Mohseni, M.; Mruczkiewicz, W.; Mutus, J.; Naaman, O.; Neeley, M.; Neill, C.; Neven, H.; Newman, M.; Niu, M. Y.; O’Brien, T. E.; Ostby, E.; Pató, B.; Petukhov, A.; Putterman, H.; Quintana, C.; Reiner, J.-M.; Roushan, P.; Rubin, N. C.; Sank, D.; Satzinger, K. J.; Smelyanskiy, V.; Strain, D.; Sung, K. J.; Schmitteckert, P.; Szalay, M.; Tubman, N. M.; Vainsencher, A.; White, T.; Vogt, N.; Yao, Z. J.; Yeh, P.; Zalcman, A.; Zanker, S. Observation of Separated Dynamics of Charge and Spin in the Fermi-Hubbard Model. arXiv, 2010.07965 [quant-ph], October 2020, http://arxiv.org/abs/2010.07965 (accessed 2023-01-10).
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    Temme, K.; Bravyi, S.; Gambetta, J. M. Error Mitigation for Short-Depth Quantum Circuits. Phys. Rev. Lett. 2017, 119 (18), 180509,  DOI: 10.1103/PhysRevLett.119.180509
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    23
    Error mitigation for short-depth quantum circuits
    Temme, Kristan; Bravyi, Sergey; Gambetta, Jay M.
    Physical Review Letters (2017), 119 (18), 180509/1-180509/5CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
    Two schemes are presented that mitigate the effect of errors and decoherence in short-depth quantum circuits. The size of the circuits for which these techniques can be applied is limited by the rate at which the errors in the computation are introduced. Near-term applications of early quantum devices, such as quantum simulations, rely on accurate ests. of expectation values to become relevant. Decoherence and gate errors lead to wrong ests. of the expectation values of observables used to evaluate the noisy circuit. The two schemes we discuss are deliberately simple and do not require addnl. qubit resources, so to be as practically relevant in current expts. as possible. The first method, extrapolation to the zero noise limit, subsequently cancels powers of the noise perturbations by an application of Richardson's deferred approach to the limit. The second method cancels errors by resampling randomized circuits according to a quasiprobability distribution.
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    McClean, J. R.; Kimchi-Schwartz, M. E.; Carter, J.; de Jong, W. A. Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States. Phys. Rev. A (Coll Park) 2017, 95 (4), 042308,  DOI: 10.1103/PhysRevA.95.042308
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    Aliferis, P.; Gottesman, D.; Preskill, J. Accuracy Threshold for Postselected Quantum Computation. arXiv, quant-ph/0703264, September 2007, http://arxiv.org/abs/quant-ph/0703264 (accessed 2023-01-09).
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    Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J. C.; Barends, R.; Boixo, S.; Broughton, M.; Buckley, B. B.; Buell, D. A.; Burkett, B.; Bushnell, N.; Chen, Y.; Chen, Z.; Chiaro, B.; Collins, R.; Courtney, W.; Demura, S.; Dunsworth, A.; Farhi, E.; Fowler, A.; Foxen, B.; Gidney, C.; Giustina, M.; Graff, R.; Habegger, S.; Harrigan, M. P.; Ho, A.; Hong, S.; Huang, T.; Huggins, W. J.; Ioffe, L.; Isakov, S. v; Jeffrey, E.; Jiang, Z.; Jones, C.; Kafri, D.; Kechedzhi, K.; Kelly, J.; Kim, S.; Klimov, P. v; Korotkov, A.; Kostritsa, F.; Landhuis, D.; Laptev, P.; Lindmark, M.; Lucero, E.; Martin, O.; Martinis, J. M.; McClean, J. R.; McEwen, M.; Megrant, A.; Mi, X.; Mohseni, M.; Mruczkiewicz, W.; Mutus, J.; Naaman, O.; Neeley, M.; Neill, C.; Neven, H.; Niu, M. Y.; O’Brien, T. E.; Ostby, E.; Petukhov, A.; Putterman, H.; Quintana, C.; Roushan, P.; Rubin, N. C.; Sank, D.; Satzinger, K. J.; Smelyanskiy, V.; Strain, D.; Sung, K. J.; Szalay, M.; Takeshita, T. Y.; Vainsencher, A.; White, T.; Wiebe, N.; Yao, Z. J.; Yeh, P.; Zalcman, A. Hartree-Fock on a Superconducting Qubit Quantum Computer. Science 2020, 369 (6507), 1084– 1089,  DOI: 10.1126/science.abb9811
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    Hartree-Fock on a superconducting qubit quantum computer
    Arute, Frank; Arya, Kunal; Babbush, Ryan; Bacon, Dave; Bardin, Joseph C.; Barends, Rami; Boixo, Sergio; Broughton, Michael; Buckley, Bob B.; Buell, David A.; Burkett, Brian; Bushnell, Nicholas; Chen, Yu; Chen, Zijun; Chiaro, Benjamin; Collins, Roberto; Courtney, William; Demura, Sean; Dunsworth, Andrew; Farhi, Edward; Fowler, Austin; Foxen, Brooks; Gidney, Craig; Giustina, Marissa; Graff, Rob; Habegger, Steve; Harrigan, Matthew P.; Ho, Alan; Hong, Sabrina; Huang, Trent; Huggins, William J.; Ioffe, Lev; Isakov, Sergei V.; Jeffrey, Evan; Jiang, Zhang; Jones, Cody; Kafri, Dvir; Kechedzhi, Kostyantyn; Kelly, Julian; Kim, Seon; Klimov, Paul V.; Korotkov, Alexander; Kostritsa, Fedor; Landhuis, David; Laptev, Pavel; Lindmark, Mike; Lucero, Erik; Martin, Orion; Martinis, John M.; McClean, Jarrod R.; McEwen, Matt; Megrant, Anthony; Mi, Xiao; Mohseni, Masoud; Mruczkiewicz, Wojciech; Mutus, Josh; Naaman, Ofer; Neeley, Matthew; Neill, Charles; Neven, Hartmut; Niu, Murphy Yuezhen; O'Brien, Thomas E.; Ostby, Eric; Petukhov, Andre; Putterman, Harald; Quintana, Chris; Roushan, Pedram; Rubin, Nicholas C.; Sank, Daniel; Satzinger, Kevin J.; Smelyanskiy, Vadim; Strain, Doug; Sung, Kevin J.; Szalay, Marco; Takeshita, Tyler Y.; Vainsencher, Amit; White, Theodore; Wiebe, Nathan; Yao, Z. Jamie; Yeh, Ping; Zalcman, Adam
    Science (Washington, DC, United States) (2020), 369 (6507), 1084-1089CODEN: SCIEAS; ISSN:1095-9203. (American Association for the Advancement of Science)
    The simulation of fermionic systems is among the most anticipated applications of quantum computing. We performed several quantum simulations of chem. with up to one dozen qubits, including modeling the isomerization mechanism of diazene. We also demonstrated error-mitigation strategies based on N-representability that dramatically improve the effective fidelity of our expts. Our parameterized ansatz circuits realized the Givens rotation approach to noninteracting fermion evolution, which we variationally optimized to prep. the Hartree-Fock wave function. This ubiquitous algorithmic primitive is classically tractable to simulate yet still generates highly entangled states over the computational basis, which allowed us to assess the performance of our hardware and establish a foundation for scaling up correlated quantum chem. simulations.
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    Huggins, W. J.; McArdle, S.; O’Brien, T. E.; Lee, J.; Rubin, N. C.; Boixo, S.; Whaley, K. B.; Babbush, R.; McClean, J. R. Virtual Distillation for Quantum Error Mitigation. Phys. Rev. X 2021, 11 (4), 041036,  DOI: 10.1103/PhysRevX.11.041036
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    27
    Virtual Distillation for Quantum Error Mitigation
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    Physical Review X (2021), 11 (4), 041036CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
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  • Abstract

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    Figure 1

    Figure 1. (a) One-dimensional representation of the electronic energy E as a function of quantum circuit parameters θ⃗. The REM approach can be explained as a four-step process: (1) A computationally tractable reference solution (such as Hartree–Fock) is computed on a classical computer. (2) A quantum measurement of the VQE surface (orange) is made using parameters corresponding to the reference solution. (3) The difference in energy calculated for the reference solution on classical and quantum hardware defines an error, ΔEREM. (4) The error estimate is assumed to be systematic and is used to correct the VQE surface in the proximity of the reference coordinate. The resulting REM corrected surface and the exact (noise-free) solution are represented by solid green and dashed red lines, respectively. Blue dots represent the coordinates of the reference calculations, while diamonds indicate minima of the different energy landscapes. (b) The inset shows the remaining error after application of REM. A possible difference in the location of the minima is indicated by θ⃗-shift (b). The difference in energy between the VQE and REM minima is indicated by ΔEerror,REM.

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    Figure 2

    Figure 2. Top: Potential energy surfaces for the dissociation of H2 and HeH+. Exact noise-free solutions from state-vector simulations are represented by black lines. Regular VQE energies obtained using a quantum computer are shown as blue dots. Results following readout mitigation and REM are shown as orange crosses and green squares, respectively. The combination of both readout mitigation and REM is shown as red stars. Measurements for H2 and HeH+ were performed on Chalmers Särimner and ibmq_quito, respectively. Bottom: error of the different approaches relative to the exact solution in the given minimal basis set. The gray region corresponds to an error of 1.6 millihartree (1 kcal/mol) with respect to the corresponding noise-free calculations.

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    Figure 3

    Figure 3. Absolute errors as a function of increasing depolarizing errors for ground state calculations of H2 and HeH+. The two-qubit depolarizing error of the Chalmers Särimner device is indicated by a vertical line for reference. Bottom inset: Energy errors after application of REM are largely independent of the noise level, and the computational accuracy is consistently close to or below 1.6 millihartree (1 kcal/mol) for these molecules.

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      Peruzzo, Alberto; McClean, Jarrod; Shadbolt, Peter; Yung, Man-Hong; Zhou, Xiao-Qi; Love, Peter J.; Aspuru-Guzik, Alan; O'Brien, Jeremy L.
      Nature Communications (2014), 5 (), 4213CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
      Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the phys. dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estn. algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state prepn. based on ansatze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We exptl. demonstrate the feasibility of this approach with an example from quantum chem.-calcg. the ground-state mol. energy for He-H+. The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future.
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    10. 10
      Farhi, E.; Goldstone, J.; Gutmann, S. A Quantum Approximate Optimization Algorithm. arXiv , 1411.4028 [quant-ph], November 14, 2014, https://arxiv.org/abs/1411.4028 (accessed 2023-01-10).
      There is no corresponding record for this reference.
    11. 11
      Preskill, J. Quantum Computing in the NISQ Era and Beyond. Quantum 2018, 2 (July), 79,  DOI: 10.22331/q-2018-08-06-79
      There is no corresponding record for this reference.
    12. 12
      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 (3), 031022,  DOI: 10.1103/PhysRevX.8.031022
      12
      Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator
      Hempel, Cornelius; Maier, Christine; Romero, Jonathan; McClean, Jarrod; Monz, Thomas; Shen, Heng; Jurcevic, Petar; Lanyon, Ben P.; Love, Peter; Babbush, Ryan; Aspuru-Guzik, Alan; Blatt, Rainer; Roos, Christian F.
      Physical Review X (2018), 8 (3), 031022CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
      Quantum-classical hybrid algorithms are emerging as promising candidates for near-term practical applications of quantum information processors in a wide variety of fields ranging from chem. to physics and materials science. We report on the exptl. implementation of such an algorithm to solve a quantum chem. problem, using a digital quantum simulator based on trapped ions. Specifically, we implement the variational quantum eigensolver algorithm to calc. the mol. ground-state energies of two simple mols. and exptl. demonstrate and compare different encoding methods using up to four qubits. Furthermore, we discuss the impact of measurement noise as well as mitigation strategies and indicate the potential for adaptive implementations focused on reaching chem. accuracy, which may serve as a cross-platform benchmark for multiqubit quantum simulators.
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    13. 13
      Sawaya, N. P. D.; Smelyanskiy, M.; McClean, J. R.; Aspuru-Guzik, A. Error Sensitivity to Environmental Noise in Quantum Circuits for Chemical State Preparation. J. Chem. Theory Comput 2016, 12 (7), 3097– 3108,  DOI: 10.1021/acs.jctc.6b00220
      13
      Error Sensitivity to Environmental Noise in Quantum Circuits for Chemical State Preparation
      Sawaya, Nicolas P. D.; Smelyanskiy, Mikhail; McClean, Jarrod R.; Aspuru-Guzik, Alan
      Journal of Chemical Theory and Computation (2016), 12 (7), 3097-3108CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Calcg. mol. energies is likely to be one of the first useful applications to achieve quantum supremacy, performing faster on a quantum than a classical computer. However, if future quantum devices are to produce accurate calcns., errors due to environmental noise and algorithmic approxns. need to be characterized and reduced. In this study, we use the high performance qHiPSTER software to investigate the effects of environmental noise on the prepn. of quantum chem. states. We simulated 18 16-qubit quantum circuits under environmental noise, each corresponding to a unitary coupled cluster state prepn. of a different mol. or mol. configuration. Addnl., we analyze the nature of simple gate errors in noise-free circuits of up to 40 qubits. We find that, in most cases, the Jordan-Wigner (JW) encoding produces smaller errors under a noisy environment as compared to the Bravyi-Kitaev (BK) encoding. For the JW encoding, pure dephasing noise is shown to produce substantially smaller errors than pure relaxation noise of the same magnitude. We report error trends in both mol. energy and electron particle no. within a unitary coupled cluster state prepn. scheme, against changes in nuclear charge, bond length, no. of electrons, noise types, and noise magnitude. These trends may prove to be useful in making algorithmic and hardware-related choices for quantum simulation of mol. energies.
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    14. 14
      Piveteau, C.; Sutter, D.; Bravyi, S.; Gambetta, J. M.; Temme, K. Error Mitigation for Universal Gates on Encoded Qubits. Phys. Rev. Lett. 2021, 127 (20), 200505,  DOI: 10.1103/PhysRevLett.127.200505
      14
      Error Mitigation for Universal Gates on Encoded Qubits
      Piveteau, Christophe; Sutter, David; Bravyi, Sergey; Gambetta, Jay M.; Temme, Kristan
      Physical Review Letters (2021), 127 (20), 200505CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
      The Eastin-Knill theorem states that no quantum error-correcting code can have a universal set of transversal gates. For Calderbank-Shor-Steane codes that can implement Clifford gates transversally, it suffices to provide one addnl. non-Clifford gate, such as the T gate, to achieve universality. Common methods to implement fault-tolerant T gates, e.g., magic state distn., generate a significant hardware overhead that will likely prevent their practical usage in the near-term future. Recently, methods have been developed to mitigate the effect of noise in shallow quantum circuits that are not protected by error correction. Error mitigation methods require no addnl. hardware resources but suffer from a bad asymptotic scaling and apply only to a restricted class of quantum algorithms. In this Letter, we combine both approaches and show how to implement encoded Clifford+T circuits where Clifford gates are protected from noise by error correction while errors introduced by noisy encoded T gates are mitigated using the quasiprobability method. As a result, Clifford+T circuits with a no. of T gates inversely proportional to the phys. noise rate can be implemented on small error-cor. devices without magic state distn. We argue that such circuits can be out of reach for state-of-the-art classical simulation algorithms.
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    15. 15
      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 (7671), 242– 246,  DOI: 10.1038/nature23879
      15
      Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets
      Kandala, Abhinav; Mezzacapo, Antonio; Temme, Kristan; Takita, Maika; Brink, Markus; Chow, Jerry M.; Gambetta, Jay M.
      Nature (London, United Kingdom) (2017), 549 (7671), 242-246CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
      Quantum computers can be used to address electronic-structure problems and problems in materials science and condensed matter physics that can be formulated as interacting fermionic problems, problems which stretch the limits of existing high-performance computers. Finding exact solns. to such problems numerically has a computational cost that scales exponentially with the size of the system, and Monte Carlo methods are unsuitable owing to the fermionic sign problem. These limitations of classical computational methods have made solving even few-atom electronic-structure problems interesting for implementation using medium-sized quantum computers. Yet exptl. implementations have so far been restricted to mols. involving only hydrogen and helium. Here we demonstrate the exptl. optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms, detg. the ground-state energy for mols. of increasing size, up to BeH2. We achieve this result by using a variational quantum eigenvalue solver (eigensolver) with efficiently prepd. trial states that are tailored specifically to the interactions that are available in our quantum processor, combined with a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine. We demonstrate the flexibility of our approach by applying it to a problem of quantum magnetism, an antiferromagnetic Heisenberg model in an external magnetic field. In all cases, we find agreement between our expts. and numerical simulations using a model of the device with noise. Our results help to elucidate the requirements for scaling the method to larger systems and for bridging the gap between key problems in high-performance computing and their implementation on quantum hardware.
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    16. 16
      Suzuki, Y.; Endo, S.; Fujii, K.; Tokunaga, Y. Quantum Error Mitigation as a Universal Error Reduction Technique: Applications from the NISQ to the Fault-Tolerant Quantum Computing Eras. PRX Quantum 2022, 3 (1), 010345  DOI: 10.1103/PRXQuantum.3.010345
      There is no corresponding record for this reference.
    17. 17
      Bravyi, S.; Sheldon, S.; Kandala, A.; McKay, D. C.; Gambetta, J. M. Mitigating Measurement Errors in Multiqubit Experiments. Phys. Rev. A (Coll Park) 2021, 103 (4), 042605,  DOI: 10.1103/PhysRevA.103.042605
      There is no corresponding record for this reference.
    18. 18
      Li, Y.; Benjamin, S. C. Efficient Variational Quantum Simulator Incorporating Active Error Minimization. Phys. Rev. X 2017, 7 (2), 021050,  DOI: 10.1103/PhysRevX.7.021050
      18
      Efficient variational quantum simulator incorporating active error minimization
      Li, Ying; Benjamin, Simon C.
      Physical Review X (2017), 7 (2), 021050/1-021050/14CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
      One of the key applications for quantum computers will be the simulation of other quantum systems that arise in chem., materials science, etc., in order to accelerate the process of discovery. It is important to ask the following question: Can this simulation be achieved using near-future quantum processors, of modest size and under imperfect control, or must it await the more distant era of large-scale fault-tolerant quantum computing. Here, we propose a variational method involving closely integrated classical and quantum coprocessors. We presume that all operations in the quantum coprocessor are prone to error. The impact of such errors is minimized by boosting them artificially and then extrapolating to the zero-error case. In comparison to a more conventional optimized Trotterization technique, we find that our protocol is efficient and appears to be fundamentally more robust against error accumulation.
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    19. 19
      Giurgica-Tiron, T.; Hindy, Y.; Larose, R.; Mari, A.; Zeng, W. J. Digital Zero Noise Extrapolation for Quantum Error Mitigation. Proceedings - IEEE International Conference on Quantum Computing and Engineering, QCE 2020; IEEE: 2020; pp 306– 316.
      There is no corresponding record for this reference.
    20. 20
      Czarnik, P.; Arrasmith, A.; Coles, P. J.; Cincio, L. Error Mitigation with Clifford Quantum-Circuit Data. Quantum 2021, 5, 592,  DOI: 10.22331/q-2021-11-26-592
      There is no corresponding record for this reference.
    21. 21
      Montanaro, A.; Stanisic, S. Error Mitigation by Training with Fermionic Linear Optics. arXiv, 2102.02120 [quant-ph], February 2021, http://arxiv.org/abs/2102.02120 (accessed 2023-01-09).
      There is no corresponding record for this reference.
    22. 22
      Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J. C.; Barends, R.; Bengtsson, A.; Boixo, S.; Broughton, M.; Buckley, B. B.; Buell, D. A.; Burkett, B.; Bushnell, N.; Chen, Y.; Chen, Z.; Chen, Y.-A.; Chiaro, B.; Collins, R.; Cotton, S. J.; Courtney, W.; Demura, S.; Derk, A.; Dunsworth, A.; Eppens, D.; Eckl, T.; Erickson, C.; Farhi, E.; Fowler, A.; Foxen, B.; Gidney, C.; Giustina, M.; Graff, R.; Gross, J. A.; Habegger, S.; Harrigan, M. P.; Ho, A.; Hong, S.; Huang, T.; Huggins, W.; Ioffe, L. B.; Isakov, S. v; Jeffrey, E.; Jiang, Z.; Jones, C.; Kafri, D.; Kechedzhi, K.; Kelly, J.; Kim, S.; Klimov, P. v; Korotkov, A. N.; Kostritsa, F.; Landhuis, D.; Laptev, P.; Lindmark, M.; Lucero, E.; Marthaler, M.; Martin, O.; Martinis, J. M.; Marusczyk, A.; McArdle, S.; McClean, J. R.; McCourt, T.; McEwen, M.; Megrant, A.; Mejuto-Zaera, C.; Mi, X.; Mohseni, M.; Mruczkiewicz, W.; Mutus, J.; Naaman, O.; Neeley, M.; Neill, C.; Neven, H.; Newman, M.; Niu, M. Y.; O’Brien, T. E.; Ostby, E.; Pató, B.; Petukhov, A.; Putterman, H.; Quintana, C.; Reiner, J.-M.; Roushan, P.; Rubin, N. C.; Sank, D.; Satzinger, K. J.; Smelyanskiy, V.; Strain, D.; Sung, K. J.; Schmitteckert, P.; Szalay, M.; Tubman, N. M.; Vainsencher, A.; White, T.; Vogt, N.; Yao, Z. J.; Yeh, P.; Zalcman, A.; Zanker, S. Observation of Separated Dynamics of Charge and Spin in the Fermi-Hubbard Model. arXiv, 2010.07965 [quant-ph], October 2020, http://arxiv.org/abs/2010.07965 (accessed 2023-01-10).
      There is no corresponding record for this reference.
    23. 23
      Temme, K.; Bravyi, S.; Gambetta, J. M. Error Mitigation for Short-Depth Quantum Circuits. Phys. Rev. Lett. 2017, 119 (18), 180509,  DOI: 10.1103/PhysRevLett.119.180509
      23
      Error mitigation for short-depth quantum circuits
      Temme, Kristan; Bravyi, Sergey; Gambetta, Jay M.
      Physical Review Letters (2017), 119 (18), 180509/1-180509/5CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
      Two schemes are presented that mitigate the effect of errors and decoherence in short-depth quantum circuits. The size of the circuits for which these techniques can be applied is limited by the rate at which the errors in the computation are introduced. Near-term applications of early quantum devices, such as quantum simulations, rely on accurate ests. of expectation values to become relevant. Decoherence and gate errors lead to wrong ests. of the expectation values of observables used to evaluate the noisy circuit. The two schemes we discuss are deliberately simple and do not require addnl. qubit resources, so to be as practically relevant in current expts. as possible. The first method, extrapolation to the zero noise limit, subsequently cancels powers of the noise perturbations by an application of Richardson's deferred approach to the limit. The second method cancels errors by resampling randomized circuits according to a quasiprobability distribution.
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    24. 24
      McClean, J. R.; Kimchi-Schwartz, M. E.; Carter, J.; de Jong, W. A. Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States. Phys. Rev. A (Coll Park) 2017, 95 (4), 042308,  DOI: 10.1103/PhysRevA.95.042308
      There is no corresponding record for this reference.
    25. 25
      Aliferis, P.; Gottesman, D.; Preskill, J. Accuracy Threshold for Postselected Quantum Computation. arXiv, quant-ph/0703264, September 2007, http://arxiv.org/abs/quant-ph/0703264 (accessed 2023-01-09).
      There is no corresponding record for this reference.
    26. 26
      Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J. C.; Barends, R.; Boixo, S.; Broughton, M.; Buckley, B. B.; Buell, D. A.; Burkett, B.; Bushnell, N.; Chen, Y.; Chen, Z.; Chiaro, B.; Collins, R.; Courtney, W.; Demura, S.; Dunsworth, A.; Farhi, E.; Fowler, A.; Foxen, B.; Gidney, C.; Giustina, M.; Graff, R.; Habegger, S.; Harrigan, M. P.; Ho, A.; Hong, S.; Huang, T.; Huggins, W. J.; Ioffe, L.; Isakov, S. v; Jeffrey, E.; Jiang, Z.; Jones, C.; Kafri, D.; Kechedzhi, K.; Kelly, J.; Kim, S.; Klimov, P. v; Korotkov, A.; Kostritsa, F.; Landhuis, D.; Laptev, P.; Lindmark, M.; Lucero, E.; Martin, O.; Martinis, J. M.; McClean, J. R.; McEwen, M.; Megrant, A.; Mi, X.; Mohseni, M.; Mruczkiewicz, W.; Mutus, J.; Naaman, O.; Neeley, M.; Neill, C.; Neven, H.; Niu, M. Y.; O’Brien, T. E.; Ostby, E.; Petukhov, A.; Putterman, H.; Quintana, C.; Roushan, P.; Rubin, N. C.; Sank, D.; Satzinger, K. J.; Smelyanskiy, V.; Strain, D.; Sung, K. J.; Szalay, M.; Takeshita, T. Y.; Vainsencher, A.; White, T.; Wiebe, N.; Yao, Z. J.; Yeh, P.; Zalcman, A. Hartree-Fock on a Superconducting Qubit Quantum Computer. Science 2020, 369 (6507), 1084– 1089,  DOI: 10.1126/science.abb9811
      26
      Hartree-Fock on a superconducting qubit quantum computer
      Arute, Frank; Arya, Kunal; Babbush, Ryan; Bacon, Dave; Bardin, Joseph C.; Barends, Rami; Boixo, Sergio; Broughton, Michael; Buckley, Bob B.; Buell, David A.; Burkett, Brian; Bushnell, Nicholas; Chen, Yu; Chen, Zijun; Chiaro, Benjamin; Collins, Roberto; Courtney, William; Demura, Sean; Dunsworth, Andrew; Farhi, Edward; Fowler, Austin; Foxen, Brooks; Gidney, Craig; Giustina, Marissa; Graff, Rob; Habegger, Steve; Harrigan, Matthew P.; Ho, Alan; Hong, Sabrina; Huang, Trent; Huggins, William J.; Ioffe, Lev; Isakov, Sergei V.; Jeffrey, Evan; Jiang, Zhang; Jones, Cody; Kafri, Dvir; Kechedzhi, Kostyantyn; Kelly, Julian; Kim, Seon; Klimov, Paul V.; Korotkov, Alexander; Kostritsa, Fedor; Landhuis, David; Laptev, Pavel; Lindmark, Mike; Lucero, Erik; Martin, Orion; Martinis, John M.; McClean, Jarrod R.; McEwen, Matt; Megrant, Anthony; Mi, Xiao; Mohseni, Masoud; Mruczkiewicz, Wojciech; Mutus, Josh; Naaman, Ofer; Neeley, Matthew; Neill, Charles; Neven, Hartmut; Niu, Murphy Yuezhen; O'Brien, Thomas E.; Ostby, Eric; Petukhov, Andre; Putterman, Harald; Quintana, Chris; Roushan, Pedram; Rubin, Nicholas C.; Sank, Daniel; Satzinger, Kevin J.; Smelyanskiy, Vadim; Strain, Doug; Sung, Kevin J.; Szalay, Marco; Takeshita, Tyler Y.; Vainsencher, Amit; White, Theodore; Wiebe, Nathan; Yao, Z. Jamie; Yeh, Ping; Zalcman, Adam
      Science (Washington, DC, United States) (2020), 369 (6507), 1084-1089CODEN: SCIEAS; ISSN:1095-9203. (American Association for the Advancement of Science)
      The simulation of fermionic systems is among the most anticipated applications of quantum computing. We performed several quantum simulations of chem. with up to one dozen qubits, including modeling the isomerization mechanism of diazene. We also demonstrated error-mitigation strategies based on N-representability that dramatically improve the effective fidelity of our expts. Our parameterized ansatz circuits realized the Givens rotation approach to noninteracting fermion evolution, which we variationally optimized to prep. the Hartree-Fock wave function. This ubiquitous algorithmic primitive is classically tractable to simulate yet still generates highly entangled states over the computational basis, which allowed us to assess the performance of our hardware and establish a foundation for scaling up correlated quantum chem. simulations.
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    27. 27
      Huggins, W. J.; McArdle, S.; O’Brien, T. E.; Lee, J.; Rubin, N. C.; Boixo, S.; Whaley, K. B.; Babbush, R.; McClean, J. R. Virtual Distillation for Quantum Error Mitigation. Phys. Rev. X 2021, 11 (4), 041036,  DOI: 10.1103/PhysRevX.11.041036
      27
      Virtual Distillation for Quantum Error Mitigation
      Huggins, William J.; McArdle, Sam; O'Brien, Thomas E.; Lee, Joonho; Rubin, Nicholas C.; Boixo, Sergio; Whaley, K. Birgitta; Babbush, Ryan; McClean, Jarrod R.
      Physical Review X (2021), 11 (4), 041036CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
      Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calcns., even with a large no. of qubits. Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then, it will be necessary to use alternative strategies to mitigate the impact of errors. We propose a near-term friendly strategy to mitigate errors by entangling and measuring M copies of a noisy state ρ. This enables us to est. expectation values with respect to a state with dramatically reduced error ρM/Tr(ρM) without explicitly prepg. it, hence the name "virtual distn." As M increases, this state approaches the closest pure state to ρ exponentially quickly. We analyze the effectiveness of virtual distn. and find that it is governed in many regimes by the behavior of this pure state (corresponding to the dominant eigenvector of ρ). We numerically demonstrate that virtual distn. is capable of suppressing errors by multiple orders of magnitude and explain how this effect is enhanced as the system size grows. Finally, we show that this technique can improve the convergence of randomized quantum algorithms, even in the absence of device noise.
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    28. 28
      Bonet-Monroig, X.; Sagastizabal, R.; Singh, M.; O’Brien, T. E. Low-Cost Error Mitigation by Symmetry Verification. Phys. Rev. A 2018, 98 (6), 062339,  DOI: 10.1103/PhysRevA.98.062339
      28
      Low-cost error mitigation by symmetry verification
      Bonet-Monroig, X.; Sagastizabal, R.; Singh, M.; O'Brien, T. E.
      Physical Review A (2018), 98 (6), 062339CODEN: PRAHC3; ISSN:2469-9934. (American Physical Society)
      A review. We investigate the performance of error mitigation via measurement of conserved symmetries on near-term devices. We present two protocols to measure conserved symmetries during the bulk of an expt., and develop a third, zero-cost, post-processing protocol which is equiv. to a variant of the quantum subspace expansion. We develop methods for inserting global and local symmetries into quantum algorithms, and for adjusting natural symmetries of the problem to boost the mitigation of errors produced by different noise channels. We demonstrate these techniques on two- and four-qubit simulations of the hydrogen mol. (using a classical d.-matrix simulator), finding up to an order of magnitude redn. of the error in obtaining the ground-state dissocn. curve.
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    29. 29
      McCaskey, A. J.; Parks, Z. P.; Jakowski, J.; Moore, S. v.; Morris, T. D.; Humble, T. S.; Pooser, R. C. Quantum Chemistry as a Benchmark for Near-Term Quantum Computers. npj Quantum Inf 2019, 5 (1), 99,  DOI: 10.1038/s41534-019-0209-0
      There is no corresponding record for this reference.
    30. 30
      McClean, J. R.; Romero, J.; Babbush, R.; Aspuru-Guzik, A. The Theory of Variational Hybrid Quantum-Classical Algorithms. New J. Phys. 2016, 18 (2), 023023,  DOI: 10.1088/1367-2630/18/2/023023
      30
      The theory of variational hybrid quantum-classical algorithms
      McClean, Jarrod R.; Romero, Jonathan; Babbush, Ryan; Aspuru-Guzik, Alan
      New Journal of Physics (2016), 18 (Feb.), 023023/1-023023/22CODEN: NJOPFM; ISSN:1367-2630. (IOP Publishing Ltd.)
      A review. Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as 'the quantum variational eigensolver' was developed with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through a relaxation of exponential operator splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Addnl., we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern deriv. free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.
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    31. 31
      Hoffmann, R.; Schleyer, P. V. R.; Schaefer, H. F. Predicting Molecules - More Realism, Please!. Angewandte Chemie - International Edition 2008, 47 (38), 7164– 7167,  DOI: 10.1002/anie.200801206
      31
      Predicting molecules--more realism, please!
      Hoffmann Roald; Schleyer Paul von Rague; Schaefer Henry F 3rd
      Angewandte Chemie (International ed. in English) (2008), 47 (38), 7164-7 ISSN:.
      There is no expanded citation for this reference.
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    32. 32
      Elfving, V. E.; Broer, B. W.; Webber, M.; Gavartin, J.; Halls, M. D.; Lorton, K. P.; Bochevarov, A. How Will Quantum Computers Provide an Industrially Relevant Computational Advantage in Quantum Chemistry? arXiv, 2009.12472, September 2020, http://arxiv.org/abs/2009.12472 (accessed 2023-01-10).
      There is no corresponding record for this reference.
    33. 33
      O’Malley, P. J. J.; Babbush, R.; Kivlichan, I. D.; Romero, J.; McClean, J. R.; Barends, R.; Kelly, J.; Roushan, P.; Tranter, A.; Ding, N.; Campbell, B.; Chen, Y.; Chen, Z.; Chiaro, B.; Dunsworth, A.; Fowler, A. G.; Jeffrey, E.; Lucero, E.; Megrant, A.; Mutus, J. Y.; Neeley, M.; Neill, C.; Quintana, C.; Sank, D.; Vainsencher, A.; Wenner, J.; White, T. C.; Coveney, P. V.; Love, P. J.; Neven, H.; Aspuru-Guzik, A.; Martinis, J. M. Scalable Quantum Simulation of Molecular Energies. Phys. Rev. X 2016, 6 (3), 031007,  DOI: 10.1103/PhysRevX.6.031007
      33
      Scalable quantum simulation of molecular energies
      O'Malley, P. J. J.; Babbush, R.; Kivlichan, I. D.; Romero, J.; McClean, J. R.; Barends, R.; Kelly, J.; Roushan, P.; Tranter, A.; Ding, N.; Campbell, B.; Chen, Y.; Chen, Z.; Chiaro, B.; Dunsworth, A.; Fowler, A. G.; Jeffrey, E.; Lucero, E.; Megrant, A.; Mutus, J. Y.; Neeley, M.; Neill, C.; Quintana, C.; Sank, D.; Vainsencher, A.; Wenner, J.; White, T. C.; Coveney, P. V.; Love, P. J.; Neven, H.; Aspuru-Guzik, A.; Martinis, J. M.
      Physical Review X (2016), 6 (3), 031007/1-031007/13CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
      We report the first electronic structure calcn. performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of mol. hydrogen using two distinct quantum algorithms. First, we exptl. execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissocn. energy to within chem. accuracy of the numerically exact result. Second, we exptl. demonstrate the canonical quantum algorithm for chem., which consists of Trotterization and quantum phase estn. We compare the exptl. performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable mols. may be viable in the near future.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhslyrtbvF&md5=750ce9fcb52cba1233a0b592d09142e3
    34. 34
      Köppl, C.; Werner, H.-J. Parallel and Low-Order Scaling Implementation of Hartree–Fock Exchange Using Local Density Fitting. J. Chem. Theory Comput 2016, 12 (7), 3122– 3134,  DOI: 10.1021/acs.jctc.6b00251
      34
      Parallel and Low-Order Scaling Implementation of Hartree-Fock Exchange Using Local Density Fitting
      Koppl Christoph; Werner Hans-Joachim
      Journal of chemical theory and computation (2016), 12 (7), 3122-34 ISSN:.
      Calculations using modern linear-scaling electron-correlation methods are often much faster than the necessary reference Hartree-Fock (HF) calculations. We report a newly implemented HF program that speeds up the most time-consuming step, namely, the evaluation of the exchange contributions to the Fock matrix. Using localized orbitals and their sparsity, local density fitting (LDF), and atomic orbital domains, we demonstrate that the calculation of the exchange matrix scales asymptotically linearly with molecular size. The remaining parts of the HF calculation scale cubically but become dominant only for very large molecular sizes or with many processing cores. The method is well parallelized, and the speedup scales well with up to about 100 CPU cores on multiple compute nodes. The effect of the local approximations on the accuracy of computed HF and local second-order Moller-Plesset perturbation theory energies is systematically investigated, and default values are established for the parameters that determine the domain sizes. Using these values, calculations for molecules with hundreds of atoms in combination with triple-ζ basis sets can be carried out in less than 1 h, with just a few compute nodes. The method can also be used to speed up density functional theory calculations with hybrid functionals that contain HF exchange.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2s%252Fnt1Wksg%253D%253D&md5=b3bdd622dd764ccb35bfcef7e0d2efbb
    35. 35
      Taube, A. G.; Bartlett, R. J. New Perspectives on Unitary Coupled-Cluster Theory. Int. J. Quantum Chem. 2006, 106 (15), 3393– 3401,  DOI: 10.1002/qua.21198
      35
      New perspectives on unitary coupled-cluster theory
      Taube, Andrew G.; Bartlett, Rodney J.
      International Journal of Quantum Chemistry (2006), 106 (15), 3393-3401CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
      The advantages and possibilities of a unitary coupled-cluster (CC) theory are examd. It is shown that using a unitary parameterization of the wave function guarantees agreement between a sum-over-states polarization propagator and response theory calcn. of properties of arbitrary order, as opposed to the case in conventional CC theory. Then, using Zassenhaus expansion for noncommuting exponential operators, explicit diagrams for an extensive and variational method based on unitary CC theory are derived. Possible extensions to the approxns. developed are discussed as well.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhtFCgsL7F&md5=715b95adcfe467301f5a54d660f4042d
    36. 36
      Johnson, R. D., III NIST Computational Chemistry Comparison and Benchmark Database. NIST Standard Reference Database Number 101. http://cccbdb.nist.gov/ (accessed 2022-11-22).
      There is no corresponding record for this reference.

  • Supporting Information

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    • Details of the Chalmers device; description of read out error mitigation; details for H2, HeH+, LiH, and BeH2 calculations, including quantum circuits and Hamiltonians; simulation details for LiH and BeH2; calibration details for ibmq_quito device; and description of depolarizing noise model used for simulations (PDF)


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