Measurement Error Mitigation for Variational Quantum Algorithms
MMeasurement Error Mitigation for Variational Quantum Algorithms
George S. Barron ∗ and Christopher J. Wood † Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA (Dated: October 19, 2020)Variational Quantum Algorithms (VQAs) are a promising application for near-term quantumprocessors, however the quality of their results is greatly limited by noise. For this reason, variouserror mitigation techniques have emerged to deal with noise that can be applied to these algorithms.Recent work introduced a technique for mitigating expectation values against correlated measure-ment errors that can be applied to measurements of 10s of qubits. We apply these techniques toVQAs and demonstrate its effectiveness in improving estimates to the cost function. Moreover, weuse the data resulting from this technique to experimentally characterize measurement errors interms of the device connectivity on devices of up to 20 qubits. These results should be useful forbetter understanding the near-term potential of VQAs as well as understanding the correlations inmeasurement errors on large, near-term devices.
I. INTRODUCTION
The development of quantum computers and theirapplications have rapidly accelerated over the last fewyears. Several different hardware platforms have beenexperimentally realized at varying scales [1–4], and therehas been an increased focus on studying algorithmsand applications that can be run on these noisy near-term devices. Some of the most promising algorithmsfor near-term devices are Variational Quantum Algo-rithms (VQAs) [5–12], which use a quantum device toevaluate an objective function that is minimized usinga classical optimizer. Instances of this algorithm in-clude the Variational Quantum Eigensolver (VQE) [6–10, 13, 14] and Quantum Approximate Optimization Al-gorithm (QAOA) [5, 15, 16]. Recent process in thisfield includes, for example, improvements to the mea-surement process [17–20], selection of variational ans¨atze[21–24], and optimization. Moreover, they have beendemonstrated experimentally on a variety of physical sys-tems [9, 25–27].The ability of current experimental implementaions ofVQAs to produce accurate results is limited by noise onthe device, despite these algorithms not explicitly requir-ing error correction. All proposed platforms for quan-tum computation experience some combination of dif-ferent errors including decoherence, calibration errors,leakage, cross-talk, and measurement errors. With su-perconducting systems, cross-talk and measurement areamong the largest sources of errors [28]. Error mitiga-tion techniques have been developed for VQAs to re-duce the amount of error on near-term devices in theabsence of error correction. For example, extrapolationto the zero-noise limit [29, 30] uses pulse-level controlto mitigate expectation values against decoherence, re-quiring only a constant factor of overhead in the number ∗ [email protected] † [email protected] of circuits executed. Techniques have also been devel-oped to characterize and mitigate against measurementerrors [31–33]. Moreover, Ref. [34] analyzes the under-lying model and provides rigorous improvements to cor-rection techniques. Related techniques have also beenused in VQE experiments [9] and are implemented in theIBM Qiskit package [35]. Recently, Ref. [36] has intro-duced a readout error mitigation technique, which we willcall Continuous-Time Markov-Process Error Mitigation(CTMP-EM), that mitigates expectation values againstcorrelated measurement errors. The n -qubit calibrationprocedure for CTMP-EM requires as few as n + 2 cir-cuits to execute and O ( n ) parameters to fit. In thiswork, they used it to mitigate estimates of the fidelityof graph states using stabilizer measurements, as well asexpectation values of stabilizers with respect to Cliffordcircuits.In this paper we apply the CTMP-EM technique to ex-perimentally characterize long-range correlations in mea-surement errors, and to calibrate error mitigated mea-surements in several quantum computers. In additionto demonstrating the presence of long-range correlationsin these devices, we are also able to show that theselong-range correlations can be as strong between distantqubits as they are between neighboring qubits. More-over, rather than only considering the global minimumfor the VQE objective function, we consider the objectivefunction holistically. Evaluating the objective function atother parameter values is important for various tasks, forexample the application of the ubiquitous parameter shiftrule [37] for analytic gradient computation in variationalexperiments. For the Fermi-Hubbard model, we demon-strate that CTMP-EM can fundamentally improve theshape of the objective function for the VQE not only atits minimum, but globally as well.This article is structured as follows. In Section II we re-view VQAs and the CTMP-EM technique. In Section IIIwe demonstrate that applying CTMP-EM to the VQEalgorithm changes the shape of the objective function.In Section IV we use the calibration data from CTMP-EM to analyze the long-range correlations in readout er- a r X i v : . [ qu a n t - ph ] O c t rors on devices. We also compare several different IBMQuantum superconducting devices with the CTMP-EMcalibration data. In Section V we conclude. II. BACKGROUNDA. Variational Quantum Algorithms
A VQA is an optimization problem min θ f ( θ ) wherethe objective function f ( θ ) is evaluated using a quantumdevice within the classical optimization loop. Two exam-ples of VQAs are the Variational Quantum Eigensolver(VQE) and the Quantum Approximate Optimization Al-gorithm (QAOA). In each case, the objective functionis f ( θ ) = (cid:104) ψ ( θ ) | H | ψ ( θ ) (cid:105) for a Hamiltonian H and pa-rameterized state | ψ ( θ ) (cid:105) . In the case of a VQE, H cor-responds to the Hamiltonian of some physical system.In the case of QAOA, H corresponds to a cost func-tion with binary variables. In QAOA, the state | ψ ( θ ) (cid:105) is prepared by alternating unitary gates correspondingto evolution of the cost Hamiltonian H and some mixeroperator(s). In VQE, the state | ψ ( θ ) (cid:105) takes the form of aspecific trial-state ansatz. The objective function f ( θ ) iscomputed by performing measurements to estimate theexpectation value (cid:104) ψ ( θ ) | H | ψ ( θ ) (cid:105) on a quantum device.VQAs are appealing for near-term devices as they areagnostic to errors in state preparation so long as the trueminimum expecation value can be reached. This has beendemonstrated on different hardware platforms, with dif-ferent instances of VQAs, and varying degrees of accu-racy. For the case of VQE it has been shown that, inspecial cases, the parameters that minimize the objec-tive function are resilient to certain types of noise [38].In general, however, device noise significantly impactsthe value of the objective function at that minimum andother points in parameter space. B. Continuous Time Markov Process ErrorMitigation
Error mitigation techniques generally aim to improvethe accuracy of the results obtained from using a noisy quantum device. Typically, each technique mitigatesagainst a certain kind of error. Measurement errors area large source of error in VQA experiments on near-termdevices. Measurement errors are modeled by a stochastic assignment matrix A acting on the state before readout.The elements of the matrix A y,x = P ( y | x ) are the prob-ability of reading out the basis state y where x was pre-pared. A general n -qubit stocastic matrix has 2 n (2 n − A matrix directly, and is used to mitigateexpectation values computed with a quantum device.This is done by modeling A = e G , where G = (cid:80) i r i G i with rates 0 ≤ r i ∈ R and operators G i that generatedifferent readout errors. In particular, for multi-qubitbitstrings a , b , the readout error a → b corresponds tothe generator G i = | b (cid:105) (cid:104) a | − | a (cid:105) (cid:104) a | . For readout errors0 ↔
1, 01 ↔
10, and 11 ↔
00 on subsets of n -qubits,CTMP-EM can determine the corresponding r i with asfew as n + 2 circuits. Once r i have been determined,Algorithm 1 of Ref. [36] can be used to estimate mea-surement error mitigated expectation values. The ideaof this algorithm is to use the measurement counts col-lected from an expectation value experiment and classi-cally post-process them by simulating a Markov processthat applies A − to the resulting distribution while simul-taneously computing the expectation value of the desiredoperator. This algorithm runs in n γe γ /δ time, where n is the number of qubits, δ is the desired additive errorin the expectation value of the Pauli string considered,and γ = − max x (cid:104) x | G | x (cid:105) for bitstrings x . Ref. [36] showsexperimentally that γ ≈ . n for up to n = 20. Hence,the algorithm requires exponential post-processing time,but in practical situations this technique is applicable upto about 50 qubits.To study the effects of measurement error mitigationon VQAs we perform simulations of noisy measurementswhich simulate the device including only readout errorspresent on individual qubits and all qubit pairs. Wechoose readout error rates to reflect the ibmq boeblingen device. However for device characterization in Section IVwe use the real IBM Quantum devices. III. MITIGATING VQA OBJECTIVEFUNCTIONSA. Ground State Energy Mitigation
One of the main applications of VQE is to estimate theground state energy of a Hamiltonian by minimizing themeasured operator expectation value over the variationalparameters. Both the expectation value and gradient es-timates are particularly senstive to measurement errorswhich can greatly effect the the performance of the clas-sical optimizer that depends on these values. This makesmeasurement error mitigation essential for improving theaccuracy of VQE and other VQAs. To investigate the im-pact of CTMP-EM on variational algorithms, we choosethe Fermi-Hubbard model which describes a system ofFermions interacting on a lattice [39]. The Hamiltonian -2 -1 0 1 2 s (a.u.)-3-2-1012 E n e r g y ( a . u . ) NoiselessMitigatedUnmitigated
FIG. 1. Sweep of the objective function f through the globalminimum ( s = 0) computed several ways. Here, f corre-sponds to the energy of the Fermi-Hubbard model for thegiven state. In the “Noiseless” case, there is no error in thesimulation and no mitigation. In the “Unmitigated” case, weinclude readout error, but no mitigation. In the “Mitigated”case, we include readout error and CTMP-EM. Estimates ofthe ground state energy ( s = 0) and surrounding points aresignificantly improved by applying CTMP-EM. is H = − t (cid:88) (cid:104) j,k (cid:105) (cid:88) σ (cid:16) a † j,σ a k,σ + a † k,σ a j,σ (cid:17) + U (cid:88) k n k, ↑ n k, ↓ , (1)where t is the tunneling parameter, and U is the interac-tion parameter between Fermions on the same site, a † k,σ is the raising operator for site k with spin σ ∈ {↑ , ↓} ,and n k,σ = a † k,σ a k,σ is the number operator. TheFermionic operators are mapped to qubit operators us-ing the Bravyi-Kitaev mapping [40]. For all calculationswe will assume U = 2 t . Energies are expressed in unitsof t . We assume that the lattice is a 1-D chain coupledby nearest neighbors with periodic boundary conditions.We chose an n -qubit variational ansatz | ψ ( θ ) (cid:105) consistingof an initial state | + (cid:105) ⊗ n followed by six repititions of alayer of parameterized single-qubit Y -rotations followedby a layer of CZ gates between neighbouring qubits.In Fig. 1 we plot the objective function f ( θ + s φ )around the global minimum, where s is the parame-ter of the sweep, φ is a randomly chosen vector withdim( θ ) = dim( φ ), and θ are the parameters that glob-ally minimize f . Evaluations of f are repeated threetimes: in the absence of any noise (“Noiseless”), includ-ing measurement error without any mitigation (“Unmit-igated”), and including measurement error with CTMP-EM applied (“Mitigated”). We observe that CTMP-EMis able to significantly improve both the estimate of theground state energy, as well as objective function valuesaround the ground state. Nevertheless, arriving at thecorrect ground state depends not only on the point con-taining the ground state itself, but the objective functionas a whole. -0.50-0.250.000.25 E n e r g y E rr o r ( a . u . ) (a) 2 4 6 8Number of Qubits0.050.100.15 S t d D e v ( a . u . ) (b) UnmitigatedMitigated FIG. 2. Comparison of samples of the objective function f ( θ )for random θ and different numbers of qubits. The energy er-ror in panel (a) is the difference between the noisy (mitigatedor unmitigated) energy and the exact result. The lefthand dis-tributions are using the unmitigated objective function, andthe righthand distributions are using the objective functionmitigated with CTMP-EM. The standard deviations in panel(b) are those of the distributions in panel (a). For all systemsizes considered, adding CTMP-EM significantly improves theestimate of the objective function. We use 8192 n shots (for n qubits) to compensate for the overhead in applying CTMP-EM. The remaining source of error in the mitigated case is dueto undersampling in the number of shots required to performCTMP-EM. This emphasizes the importance of the scalingof measurements needed for CTMP-EM with the number ofqubits. B. Objective Function Sampling
In VQE the quantum computer is treated as a blackbox evaluation of the objective function f , hence it iscritical that evaluations of f are accurate. To investi-gate the effectivness of measurement error mitigation forblack box evaluations of f ( θ ) we sample points in pa-rameter space θ and compare the noiseless, unmitigated,and mitigated cases for Fermi-Hubbard models with 1,2, 3, and 4 sites (with 2, 4, 6, and 8 qubits respectively).We evaluate the energy with and without CTMP-EM forrandomly sampled parameters of the objective functionand compare the distribution of values with the noiselessresult as shown in Fig. 2.We find that the standard deviation of the error dis-tribution is significantly reduced when mitigation is ap-plied. Specifically, for 2, 4, 6, and 8 qubits respectively, − − − − G e n e r a t o r C o e ff i c i e n t s (a) Excitation Generators 1 2 3 4 5 6 7Qubit Distance(b) Decay Generators 1 2 3 4 5 6 7Qubit Distance(c) Exchange Generators FIG. 3. Histograms and quartiles of coefficients r i for various generators G i . The “Excitation” (a), “Decay” (b), and “Exchange”(c) generators are associated with readout errors 00 →
11, 11 →
00, and 01 →
10, respectively. The “Qubit Distance” is thelength of the shortest path between two qubits on ibmq boeblingen device. Higher generator coefficients correspond to higherreadout error rates. In all cases, the long-range correlations in readout error can be non-trivial. Cases where r i = 0 due toshot limitations are elided for the purposes of plotting with logarithmic axes. In panel (b) we include the connectivity for the ibmq boeblingen
20 qubit device. The vertices represent qubits, and the edges represent connections between qubits. Despitequbits being physically separated on the device, long-range correlations between qubits can occur. the standard deviation is reduced by factors of approxi-mately 7 .
46, 7 .
64, 5 .
18, and 3 .
40. This demonstrates thatin the absence of CTMP-EM, the noisy objective func-tion deviates significantly from its noiseless form whichcan greatly limit the effectivness of the VQE algorithmeven for a small number of qubits.
IV. CHARACTERIZING READOUT ERRORS
The CTMP-EM method can also be used as a char-acterization protocol for correlated measurement errorsin a quantum device, which we will demonstrate by us-ing it to characterize correlated readout errors in severalexperimental devices. One method for measurement er-ror characterization involves computing the full A matrix(which has 2 n elements) as a form of measurement to-mography [41], however this is not possible to do past asmall number of qubits. Instead we use the set of rates { r i } of the CTMP-EM generator G to characterize cor-related measurement errors. This is scalable in the sensethat it requires as few as n + 2 circuits, and G is param-eterized by the O ( n ) rates of its generator components. A. Characterizing Correlated Errors
Calibration of G in the CTMP-EM model describedin Ref. [36] is done by preparing a input set of com-putational basis states {| a i (cid:105)} , labelled by bitstrings a i ,and performing measurements in the computational ba-sis to estimate the assignment probabilities P ( x | a i ) for x = 0 , ..., n −
1. These probabilities are then processedto compute the CTMP-EM generator rates { r i } for each of the 1 and 2-qubit generator terms. The set of input la-bels a i is complete if the set of all measurement outcomescontains all 1 and 2-qubit transitions for the CTMP-EMgenerators, if only 2-qubit correlations are present. Thisrequires at least n + 2 generators, though more may beused to provide a more uniform distribution across gen-erator terms. We use the set of all bitstrings that haveHamming weight ≤
2, of which there are ( n + n + 2) / r i grouped by the qubit distance ,which we define as the shortest path between two qubitsin terms of the device connectivity.We apply this technique to the 20 qubit ibmq boeblingen device, which has a planar qubitconnectivity graph as shown inset in Fig. 3. Here, forexample, neighboring qubits have distance 1, and somepairs of qubits in the corners of the layout have distance7. The histograms of measured 2-qubit generator rates r i vs qubit distance for the ibmq boeblingen device areshown inset in Fig. 3. Here we further group the gener-ators into three types: excitation generators (00 → → ↔ Almaden, 2Q: 0 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . − − − − − − Generator Coefficients C o un t s X2, 2Q: 0 . ± . . ± . FIG. 4. Generator coefficients for 1 (blue) and 2-qubit (or-ange) readout errors on several IBM Quantum devices. Inall cases, the rates are relatively consistent. Some error ratesare calculated as r i = 0 due to limitations in the number ofshots. These cases are elided for the purposes of plotting withlogarithmic axes. expected as thermal relaxation to the ground state is asignificant contribution to measurement errors. However,it is surprising that the distribution median does not de-cay appreciably with qubit distance and is non-negligibleeven for distantly connected qubits. The excitation andexchange generators on the other hand show reductionwith qubit distance on average, however in some casescertain rates can be as large as those between neighbor-ing qubits. B. Comparing Devices with CTMP-EM Data
To illustrate the usefulness of CTMP-EM for charac-terization, we use this method to characterize readout errors on several IBM Quantum devices and comparetheir local 1-qubit and correlated 2-qubit generator co-efficients. The distribution in generator values is shownin Fig. 4. For all measured devices the 1-qubit errorrates are generally higher than the 2-qubit error rates asexpected, and all devices give relatively consistent errorrates. The devices shown range from 5-qubit to 20-qubitswith the calibrations run using the minimum number of n + 2 calibration circuits, using the maximum number ofshots available for each device. Moreover, CTMP-EM asa characterization technique does not depend on deviceconnectivity, and includes information about long-rangecorrelations. This is advantageous since we saw beforethat the long-range correlations in readout errors can benon-trivial. V. CONCLUSION
In this paper, we have shown that CTMP-EM is vi-tal for improving the performance of VQAs on near-termdevices, and that it can fundamentally change the shapeof the objective function computed by the quantum de-vice. Moreover, we demonstrate that CTMP-EM canefficiently be used to characterize long-range correlationsin readout error on near-term devices in terms of thedevice connectivity, and that these long-range correla-tions are present on current devices. Nevertheless, aninteresting topic for future work would be to expand theCTMP model to generators that act on more than twoqubits, or analyze how the calibration parameters driftover time. Additionally, it would be interesting to inves-tigate the impact on performance for QAOA Hamilto-nians. For now, our work emphasizes the importance ofusing CTMP-EM for characterization, because it only re-quires computing the terms of G , of which there are only O ( n ) many, instead of A = e G , which is dense. More-over, the calibration technique is efficient in the numberof circuits. We believe that our results will be useful tounderstanding the objective function in VQAs on near-term devices, as well as characterizing near-term devicesin terms of readout error. ACKNOWLEDGEMENTS
The authors thank Sergey Bravyi, Sophia Economou,and Sarah Sheldon for helpful discussions. We also thankthe IBM Quantum team for providing access to the de-vices used in this work. This work was done during G.S.Binternship at IBM Quantum during the summer of 2020.G.S.B. thanks the IBM Quantum team for a very enrich-ing internship experience. [1] Andrew W. Cross, Lev S. Bishop, Sarah Sheldon, Paul D.Nation, and Jay M. Gambetta, “Validating quantumcomputers using randomized model circuits,” Phys. Rev.A , 032328 (2019).[2] J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A.Moses, C. H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer,C. Ryan-Anderson, and et al., “Demonstration of theqccd trapped-ion quantum computer architecture,” arXive-prints , 2003.01293 (2020).[3] Peter J Karalekas, Nikolas A Tezak, Eric C Peterson,Colm A Ryan, Marcus P da Silva, and Robert S Smith,“A quantum-classical cloud platform optimized for vari-ational hybrid algorithms,” Quantum Science and Tech-nology , 024003 (2020).[4] Petar Jurcevic, Ali Javadi-Abhari, Lev S. Bishop,Isaac Lauer, Daniela F. Bogorin, Markus Brink, Lau-ren Capelluto, Oktay G¨unl¨uk, Toshinaro Itoko, NaokiKanazawa, Abhinav Kandala, George A. Keefe, KevinKruslich, William Landers, Eric P. Lewandowski, Dou-glas T. McClure, Giacomo Nannicini, Adinath Naras-gond, Hasan M. Nayfeh, Emily Pritchett, Mary BethRothwell, Srikanth Srinivasan, Neereja Sundaresan,Cindy Wang, Ken X. Wei, Christopher J. Wood, Jeng-Bang Yau, Eric J. Zhang, Oliver E. Dial, Jerry M. Chow,and Jay M. Gambetta, “Demonstration of quantum vol-ume 64 on a superconducting quantum computing sys-tem,” (2020), arXiv:2008.08571 [quant-ph].[5] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, “Aquantum approximate optimization algorithm,” arXivpreprint arXiv:1411.4028 (2014).[6] A. Peruzzo, J.McClean, P. Shadbolt, M.-H.Yung, X.-Q.Zhou, P.J. Love, A. Aspuru-Guzik, and J. L. O’Brien,“A variational eigenvalue solver on a photonic quantumprocessor,” Nature Commun. , 4213 (2014).[7] P. J. J. O’Malley, R. Babbush, I. D. Kivlichan,J. Romero, J. R. McClean, R. Barends, J. Kelly,P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen,Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jef-frey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley,C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wen-ner, T. C. White, P. V. Coveney, P. J. Love, H. Neven,A. Aspuru-Guzik, and J. M. Martinis, “Scalable quan-tum simulation of molecular energies,” Phys. Rev. X ,031007 (2016).[8] Jarrod R McClean, Jonathan Romero, Ryan Babbush,and Al´an Aspuru-Guzik, “The theory of variational hy-brid quantum-classical algorithms,” New J. Phys ,023023 (2016).[9] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme,Maika Takita, Markus Brink, Jerry M Chow, andJay M Gambetta, “Hardware-efficient variational quan-tum eigensolver for small molecules and quantum mag-nets,” Nature , 242–246 (2017).[10] J. I. Colless, V. V. Ramasesh, D. Dahlen, M. S. Blok,M. E. Kimchi-Schwartz, J. R. McClean, J. Carter, W. A.de Jong, and I. Siddiqi, “Computation of molecular spec-tra on a quantum processor with an error-resilient algo-rithm,” Phys. Rev. X , 011021 (2018).[11] G Pagano, A Bapat, P Becker, KS Collins, A De,PW Hess, HB Kaplan, A Kyprianidis, WL Tan,C Baldwin, et al. , “Quantum approximate optimization with a trapped-ion quantum simulator,” arXiv preprintarXiv:1906.02700 (2019).[12] Jonathan Romero and Alan Aspuru-Guzik, “Variationalquantum generators: Generative adversarial quantummachine learning for continuous distributions,” arXivpreprint arXiv:1901.00848 (2019).[13] Pauline J Ollitrault, Abhinav Kandala, Chun-Fu Chen,Panagiotis Kl Barkoutsos, Antonio Mezzacapo, MarcoPistoia, Sarah Sheldon, Stefan Woerner, Jay Gambetta,and Ivano Tavernelli, “Quantum equation of motion forcomputing molecular excitation energies on a noisy quan-tum processor,” arXiv preprint arXiv:1910.12890 (2019),arXiv:1910.12890 [quant-ph].[14] Frank Arute, Kunal Arya, Ryan Babbush, Dave Ba-con, Joseph C. Bardin, Rami Barends, Sergio Boixo,Michael Broughton, Bob B. Buckley, David A. Buell,Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen,Benjamin Chiaro, Roberto Collins, William Courtney,Sean Demura, Andrew Dunsworth, Daniel Eppens, Ed-ward Farhi, Austin Fowler, Brooks Foxen, Craig Gid-ney, Marissa Giustina, Rob Graff, Steve Habegger,Matthew P. Harrigan, Alan Ho, Sabrina Hong, TrentHuang, William J. Huggins, Lev Ioffe, Sergei V. Isakov,Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri,Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V.Klimov, Alexander Korotkov, Fedor Kostritsa, DavidLandhuis, Pavel Laptev, Mike Lindmark, Erik Lucero,Orion Martin, John M. Martinis, Jarrod R. McClean,Matt McEwen, Anthony Megrant, Xiao Mi, MasoudMohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naa-man, Matthew Neeley, Charles Neill, Hartmut Neven,Murphy Yuezhen Niu, Thomas E. O’Brien, Eric Os-tby, Andre Petukhov, Harald Putterman, Chris Quin-tana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank,Kevin J. Satzinger, Vadim Smelyanskiy, Doug Strain,Kevin J. Sung, Marco Szalay, Tyler Y. Takeshita, AmitVainsencher, Theodore White, Nathan Wiebe, Z. JamieYao, Ping Yeh, and Adam Zalcman, “Hartree-Fockon a superconducting qubit quantum computer,” arXive-prints , arXiv:2004.04174 (2020), arXiv:2004.04174[quant-ph].[15] J. S. Otterbach, R. Manenti, N. Alidoust, A. Bestwick,M. Block, B. Bloom, S. Caldwell, N. Didier, E. SchuylerFried, S. Hong, P. Karalekas, C. B. Osborn, A. Papa-george, E. C. Peterson, G. Prawiroatmodjo, N. Rubin,Colm A. Ryan, D. Scarabelli, M. Scheer, E. A. Sete,P. Sivarajah, Robert S. Smith, A. Staley, N. Tezak, W. J.Zeng, A. Hudson, Blake R. Johnson, M. Reagor, M. P.da Silva, and C. Rigetti, “Unsupervised Machine Learn-ing on a Hybrid Quantum Computer,” arXiv e-prints ,arXiv:1712.05771 (2017), arXiv:1712.05771 [quant-ph].[16] Stuart Hadfield, Zhihui Wang, Bryan O’Gorman, EleanorRieffel, Davide Venturelli, and Rupak Biswas, “Fromthe quantum approximate optimization algorithm to aquantum alternating operator ansatz,” Algorithms ,34 (2019).[17] Ryan Babbush, Nathan Wiebe, Jarrod McClean, JamesMcClain, Hartmut Neven, and Garnet Kin-Lic Chan,“Low-depth quantum simulation of materials,” PhysicalReview X , 011044 (2018). [18] Vladyslav Verteletskyi, Tzu-Ching Yen, and Artur F.Izmaylov, “Measurement optimization in the variationalquantum eigensolver using a minimum clique cover,”The Journal of Chemical Physics , 124114 (2020),https://doi.org/10.1063/1.5141458.[19] William J. Huggins, Jarrod McClean, Nicholas Rubin,Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, andRyan Babbush, “Efficient and Noise Resilient Measure-ments for Quantum Chemistry on Near-Term QuantumComputers,” arXiv:1907.13117 [physics, physics:quant-ph] (2019), arXiv:1907.13117 [physics, physics:quant-ph].[20] Andrew Zhao, Andrew Tranter, William M. Kirby,Shu Fay Ung, Akimasa Miyake, and Peter J. Love, “Mea-surement reduction in variational quantum algorithms,”Phys. Rev. A , 062322 (2020).[21] Harper R. Grimsley, Sophia E. Economou, EdwinBarnes, and Nicholas J. Mayhall, “An adaptive vari-ational algorithm for exact molecular simulations on aquantum computer,” Nat. Commun. , 3007 (2019).[22] Panagiotis Kl. Barkoutsos, Jerome F. Gonthier, IgorSokolov, Nikolaj Moll, Gian Salis, Andreas Fuhrer, MarcGanzhorn, Daniel J. Egger, Matthias Troyer, Anto-nio Mezzacapo, Stefan Filipp, and Ivano Tavernelli,“Quantum algorithms for electronic structure calcula-tions: Particle-hole hamiltonian and optimized wave-function expansions,” Phys. Rev. A , 022322 (2018).[23] Ho Lun Tang, V. O. Shkolnikov, George S. Barron,Harper R. Grimsley, Nicholas J. Mayhall, Edwin Barnes,and Sophia E. Economou, “qubit-adapt-vqe: An adaptivealgorithm for constructing hardware-efficient ansatze ona quantum processor,” (2019), arXiv:1911.10205 [quant-ph].[24] Bryan T Gard, Linghua Zhu, George S Barron, Nicholas JMayhall, Sophia E Economou, and Edwin Barnes, “Effi-cient symmetry-preserving state preparation circuits forthe variational quantum eigensolver algorithm,” arXivpreprint arXiv:1904.10910 (2019).[25] D. Zhu, N. M. Linke, M. Benedetti, K. A. Lands-man, N. H. Nguyen, C. H. Alderete, A. Perdomo-Ortiz, N. Korda, A. Garfoot, C. Brecque, L. Egan,O. Perdomo, and C. Monroe, “Training of quan-tum circuits on a hybrid quantum computer,” Sci-ence Advances (2019), 10.1126/sciadv.aaw9918,https://advances.sciencemag.org/content/5/10/eaaw9918.full.pdf.[26] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R.Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J.Dean, and P. Lougovski, “Cloud quantum computing ofan atomic nucleus,” Phys. Rev. Lett. , 210501 (2018).[27] N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Mor-ris, R. C. Pooser, M. Sanz, E. Solano, P. Lougovski,and M. J. Savage, “Quantum-classical computation ofschwinger model dynamics using quantum computers,”Phys. Rev. A , 032331 (2018). [28] Yanzhu Chen, Maziar Farahzad, Shinjae Yoo, and Tzu-Chieh Wei, “Detector tomography on ibm quantum com-puters and mitigation of an imperfect measurement,”Phys. Rev. A , 052315 (2019).[29] Kristan Temme, Sergey Bravyi, and Jay M. Gam-betta, “Error mitigation for short-depth quantum cir-cuits,” Phys. Rev. Lett. , 180509 (2017).[30] Abhinav Kandala, Kristan Temme, Antonio D C´orcoles,Antonio Mezzacapo, Jerry M Chow, and Jay M Gam-betta, “Error mitigation extends the computational reachof a noisy quantum processor,” Nature , 491–495(2019).[31] Kathleen E. Hamilton and Raphael C. Pooser, “Error-mitigated data-driven circuit learning on noisy quan-tum hardware,” Quantum Machine Intelligence , 1–15(2020).[32] Michael R. Geller and Mingyu Sun, “Efficient correc-tion of multiqubit measurement errors,” arXiv preprintarXiv:2001.09980 (2020), arXiv:2001.09980 [quant-ph].[33] Kathleen E. Hamilton, Tyler Kharazi, Titus Mor-ris, Alexander J. McCaskey, Ryan S. Bennink, andRaphael C. Pooser, “Scalable quantum processornoise characterization,” arXiv preprint arXiv:2006.01805(2020), arXiv:2006.01805 [quant-ph].[34] Michael R Geller, “Rigorous measurement error cor-rection,” Quantum Science and Technology , 03LT01(2020).[35] https://github.com/qiskit/qiskit, “Qiskit: An open-source framework for quantum computing,” (2019).[36] Sergey Bravyi, Sarah Sheldon, Abhinav Kandala,David C. Mckay, and Jay M. Gambetta, “Mitigatingmeasurement errors in multi-qubit experiments,” arXivpreprint arXiv:2006.14044 (2020), arXiv:2006.14044[quant-ph].[37] Maria Schuld, Ville Bergholm, Christian Gogolin, JoshIzaac, and Nathan Killoran, “Evaluating analytic gra-dients on quantum hardware,” Phys. Rev. A , 032331(2019).[38] Kunal Sharma, Sumeet Khatri, M Cerezo, and Patrick JColes, “Noise resilience of variational quantum compil-ing,” New Journal of Physics , 043006 (2020).[39] J. Hubbard and Brian Hilton Flowers, “Electroncorrelations in narrow energy bands,” Proceedingsof the Royal Society of London. Series A. Math-ematical and Physical Sciences , 238–257 (1963),https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1963.0204.[40] Sergey B. Bravyi and Alexei Yu. Kitaev, “Fermionicquantum computation,” Annals of Physics , 210 –226 (2002).[41] R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero,M. Neeley, A. D. O’Connell, D. Sank, H. Wang, J. Wen-ner, M. Steffen, and et al., “Quantum process to-mography of a universal entangling gate implementedwith josephson phase qubits,” Nature Physics6