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Progress towards practical quantum variational algorithms
Open accessAbstract
The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.
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“Several QML algorithms based on PQC have been proposed [21–27], demonstrating superior performance and computational abilities compared to conventional algorithms [28–31].”
From Medical image diagnosis based on adaptive Hybrid Quantum CNN (2023) · cites this paper as [21]
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“Motivated by adiabatic state preparation, the Variational Hamiltonian Ansatz (VHA) was developed to reduce the number of parameters and accelerate the convergence (McClean et al. , 2016; Wecker et al. , 2015).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Wecker et al., 2015
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“To this end, VQE (McClean et al. , 2016; Peruzzo et al. , 2014) has been proposed, to find the ground state of a Hamiltonian H in a manner that is suited for NISQ devices (Wecker et al. , 2015).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Wecker et al. , 2015
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“The AQC model has been shown to be computationally equivalent to the gate model 62 , meaning that algorithms written in either model can be translated to ones in the other model without incurring prohibitive computational overhead.”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [62]
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Citations
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Variational quantum algorithms
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Quantum computational chemistry
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Quantum Chemistry in the Age of Quantum Computing
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., et al. (2019) · Chemical Reviews · doi:10.1021/acs.chemrev.8b00803 · Cited by 1,560
Error Mitigation for Short-Depth Quantum Circuits
Temme, K., Bravyi, S., & Gambetta, J. M. (2017) · Physical Review Letters · doi:10.1103/physrevlett.119.180509 · Cited by 1,379
Quantum Chemistry in the Age of Quantum Computing
Cao, Y., Romero, J., Olson, J., Degroote, M., Johnson, P., Kieferová, M., Kivlichan, I., Menke, T., Peropadre, B., Sawaya, N., et al. (n.d.) · OPUS - Open Publications of UTS Scholars (University of Technology Sydney) · Cited by 1,265
The Variational Quantum Eigensolver: A review of methods and best practices
Tilly, J., Chen, H., Cao, S., Picozzi, D., Setia, K., Li, Y., Grant, E., Wossnig, L., Rungger, I., Booth, G. H., et al. (2022) · Physics Reports · doi:10.1016/j.physrep.2022.08.003 · Cited by 1,031
Power of data in quantum machine learning
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Quantum Algorithms for Quantum Chemistry and Quantum Materials Science
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Scalable Quantum Simulation of Molecular Energies
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“While the unitary variant has no efficient preparation scheme on a classical computer, scalable methods of preparation for a fixed set of parameters on a quantum device have now been documented several times [19,20,22,23].”
BackgroundProgrammable quantum simulations of spin systems with trapped ions
Monroe, C. R., Campbell, W. C., Duan, L.-M., Gong, Z., Gorshkov, A. V., Hess, P., Islam, R., Kim, K., Linke, N. M., Pagano, G., et al. (2021) · Reviews of Modern Physics · doi:10.1103/revmodphys.93.025001 · Cited by 891
“Finding the classical optimization strategy and making efficient use of quantum resources is an active area of research (Cirstoiu et al., 2019; Sundar et al., 2019; Wecker et al., 2015; Yang et al., 2017).”
BackgroundProgrammable Quantum Simulations of Spin Systems with Trapped Ions
Monroe, C., Campbell, W. C., Duan, L. M., Gong, Z. X., Gorshkov, A. V., Hess, P., Islam, R., Kim, K., Linke, N., Pagano, G., et al. (2019) · arXiv · doi:10.48550/arxiv.1912.07845 · Cited by 806
Elucidating reaction mechanisms on quantum computers
Reiher, M., Wiebe, N., Svore, K. M., Wecker, D., & Troyer, M. (2017) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1619152114 · Cited by 745
“[18] Dave Wecker, Matthew B Hastings, and Matthias Troyer, “Progress towards practical quantum variational algorithms,” Physical Review A 92, 042303 (2015).”
BackgroundSelf-citationQuantum optimization using variational algorithms on near-term quantum devices
Moll, N., Barkoutsos, P., Bishop, L. S., Chow, J. M., Cross, A., Egger, D. J., Filipp, S., Fuhrer, A., Gambetta, J. M., Ganzhorn, M., et al. (2018) · Quantum Science and Technology · doi:10.1088/2058-9565/aab822 · Cited by 744
“In this context, in [38] a greedy search with multiple starting points is alternated with a Powell search, showing good performances on Hubbard lattices of up to twelve sites.”
BackgroundMethodInfluentialNoise-induced barren plateaus in variational quantum algorithms
Wang, S., Fontana, E., Cerezo, M., Sharma, K., Sone, A., Cincio, L., & Coles, P. J. (2021) · Nature Communications · doi:10.1038/s41467-021-27045-6 · Cited by 773
“[54] Dave Wecker, Matthew B Hastings, and Matthias Troyer, “Progress towards practical quantum variational algorithms,” Physical Review A 92, 042303 (2015).”
BackgroundStrategies for quantum computing molecular energies using the unitary coupled cluster ansatz
Romero, J., Babbush, R., McClean, J. R., Hempel, C., Love, P. J., & Aspuru-Guzik, A. (2019) · Quantum Science and Technology · doi:10.1088/2058-9565/aad3e4 · Cited by 726
Efficient Variational Quantum Simulator Incorporating Active Error Minimization
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“(1), and then take the evolution time of each term as a variational parameter (fλkg 1⁄4 fτn;jg) rather than a fixed value as in the Trotterization algorithm [11].”
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Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator
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“For instance, it has been argued that a parameterized Trotter approximation [7, 81], to the adiabatic state preparation algorithm may serve as an effective ansatz [40, 41, 51].”
BackgroundInfluentialSelf-verifying variational quantum simulation of lattice models
Kokail, C., Maier, C., van Bijnen, R., Brydges, T., Joshi, M. K., Jurcevic, P., Muschik, C. A., Silvi, P., Blatt, R., Roos, C. F., et al. (2019) · Nature · doi:10.1038/s41586-019-1177-4 · Cited by 594
Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus
Holmes, Z., Sharma, K., Cerezo, M., & Coles, P. J. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.010313 · Cited by 570
“For instance, the quantum alternating operator ansatz [21] (or the Hamiltonian variational ansatz [25]), encode information of an appropriate adiabatic transformation.”
BackgroundPractical Quantum Error Mitigation for Near-Future Applications
Endo, S., Benjamin, S. C., & Li, Y. (2018) · Physical Review X · doi:10.1103/physrevx.8.031027 · Cited by 562
Variational ansatz-based quantum simulation of imaginary time evolution
McArdle, S., Jones, T., Endo, S., Li, Y., Benjamin, S. C., & Yuan, X. (2019) · npj Quantum Information · doi:10.1038/s41534-019-0187-2 · Cited by 557
“This combination can make the classical optimisation step of the VQE very difficult.(43) As described above, if the initial state has a non-zero overlap with the ground state, true propagation in imaginary time will evolve the system into the ground state, in the limit that τ→∞.”
BackgroundHybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states
McClean, J. R., Kimchi-Schwartz, M. E., Carter, J., & de Jong, W. A. (2017) · Physical Review A · doi:10.1103/physreva.95.042308 · Cited by 516
Quantum autoencoders for efficient compression of quantum data
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OpenFermion: the electronic structure package for quantum computers
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Variational Quantum Computation of Excited States
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“[23] in the context of VQE, and UCCGSD has since been investigated numerically [18, 41].”
BackgroundQuantum Simulation of Electronic Structure with Linear Depth and Connectivity
Kivlichan, I. D., McClean, J., Wiebe, N., Gidney, C., Aspuru-Guzik, A., Chan, G. K.-L., & Babbush, R. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.110501 · Cited by 431
Generalized Unitary Coupled Cluster Wave functions for Quantum Computation
Lee, J., Huggins, W. J., Head-Gordon, M., & Whaley, K. B. (2019) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.8b01004 · Cited by 432
“a general twobody cluster expansion. Motivated by earlier work of Nakatsuji,33 Nooijen conjectured15 that it is possible to express an exact ground state of a two-body Hamiltonian as j i= eT^j˚ 0 i; (10) 8 where T^ = T^ 1 + T^ 2 (11) = 1 2 X pq tq p ^a y q ^a p + 1 4 X pqrs trs pq ^a y r ^a y s ^a q ^a p : (12) This y…”
ResultUniversal Quantum Control through Deep Reinforcement Learning
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Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity
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“More recently, experimental demonstrations [11 – 15] have focused on the development of variational algorithms [16,17], which are often [18,19], but not always [20,21], inspired by time-evolution primitives.”
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Self-citationLow-Depth Quantum Simulation of Materials
Babbush, R., Wiebe, N., McClean, J., McClain, J., Neven, H., & Chan, G. K.-L. (2018) · Physical Review X · doi:10.1103/physrevx.8.011044 · Cited by 335
“This ansatz is related to the quantum approximate optimization algorithm [110] and has been shown to perform well in the context of electronic structure [42].”
BackgroundMethodInfluentialLayerwise learning for quantum neural networks
Skolik, A., McClean, J. R., Mohseni, M., van der Smagt, P., & Leib, M. (2021) · Quantum Machine Intelligence · doi:10.1007/s42484-020-00036-4 · Cited by 324
Qubit Coupled Cluster Method: A Systematic Approach to Quantum Chemistry on a Quantum Computer
Ryabinkin, I. G., Yen, T.-C., Genin, S. N., & Izmaylov, A. F. (2018) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.8b00932 · Cited by 287
Commercialize quantum technologies in five years
Mohseni, M., Read, P., Neven, H., Boixo, S., Denchev, V., Babbush, R., Fowler, A., Smelyanskiy, V., & Martinis, J. (2017) · Nature · doi:10.1038/543171a · Cited by 268
Simulating Large Quantum Circuits on a Small Quantum Computer
Peng, T., Harrow, A. W., Ozols, M., & Wu, X. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.150504 · Cited by 324
Measurement optimization in the variational quantum eigensolver using a minimum clique cover
Verteletskyi, V., Yen, T.-C., & Izmaylov, A. F. (2020) · The Journal of Chemical Physics · doi:10.1063/1.5141458 · Cited by 269
Barren plateaus in variational quantum computing
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“By understanding the connections between these different origins of BPs, we will see that one should be able to predict whether a model will exhibit an exponentially concentrated loss.”
Variational quantum algorithms for discovering Hamiltonian spectra
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Quantum computing enhanced computational catalysis
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Self-citationQuantum computing enhanced computational catalysis
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Self-citationEfficient and noise resilient measurements for quantum chemistry on near-term quantum computers
Huggins, W. J., McClean, J. R., Rubin, N. C., Jiang, Z., Wiebe, N., Whaley, K. B., & Babbush, R. (2021) · npj Quantum Information · doi:10.1038/s41534-020-00341-7 · Cited by 256
Exploring Entanglement and Optimization within the Hamiltonian Variational Ansatz
Wiersema, R., Zhou, C., de Sereville, Y., Carrasquilla, J., Kim, Y. B., & Yuen, H. (2020) · PRX Quantum · doi:10.1103/prxquantum.1.020319 · Cited by 289
Error-Mitigated Digital Quantum Simulation
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Hybrid Quantum-Classical Approach to Correlated Materials
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“A different class of algorithms that have been explored over the last few years are ”variational quantum eigensolvers” [42, 43].”
BackgroundSelf-citationAccelerated Variational Quantum Eigensolver
Wang, D., Higgott, O., & Brierley, S. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.140504 · Cited by 241
Variational Quantum Simulation of General Processes
Endo, S., Sun, J., Li, Y., Benjamin, S. C., & Yuan, X. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.010501 · Cited by 237
A Practical Quantum Instruction Set ArchitecturePreprint
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Exact parameterization of fermionic wave functions via unitary coupled cluster theory
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A quantum computing view on unitary coupled cluster theory
Anand, A., Schleich, P., Alperin-Lea, S., Jensen, P. W. K., Sim, S., Díaz‐Tinoco, M., Kottmann, J. S., Degroote, M., Izmaylov, A. F., & Aspuru‐Guzik, A. (2022) · Chemical Society Reviews · doi:10.1039/d1cs00932j · Cited by 251
VQE method: a short survey and recent developments
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Diagnosing Barren Plateaus with Tools from Quantum Optimal Control
Larocca, M., Czarnik, P., Sharma, K., Muraleedharan, G., Coles, P. J., & Cerezo, M. (2022) · Quantum · doi:10.22331/q-2022-09-29-824 · Cited by 272
“We analyze the existence of barren plateaus in QAOA and HVA ansatzes, and we highlight the role of the input state, as different initial states can lead to the presence or absence of barren plateaus.”
MethodInfluentialWitnessing eigenstates for quantum simulation of Hamiltonian spectra
Santagati, R., Wang, J., Gentile, A. A., Paesani, S., Wiebe, N., McClean, J. R., Morley-Short, S., Shadbolt, P. J., Bonneau, D., Silverstone, J. W., et al. (2018) · Science Advances · doi:10.1126/sciadv.aap9646 · Cited by 211
“These advantages are significant given the high accuracy required in quantum chemical applications (24).”
BackgroundMachine Learning: Quantum vs Classical
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“Quantum chemistry simulation is considered a true application in NISQ computers [38], [39].”
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Leakage reduction in fast superconducting qubit gates via optimal control
Werninghaus, M., Egger, D. J., Roy, F., Machnes, S., Wilhelm, F. K., & Filipp, S. (2021) · npj Quantum Information · doi:10.1038/s41534-020-00346-2 · Cited by 217
Perspective on the Current State-of-the-Art of Quantum Computing for Drug Discovery Applications
Blunt, N. S., Camps, J., Crawford, O., Izsák, R., Leontica, S., Mirani, A., Moylett, A. E., Scivier, S. A., Sünderhauf, C., Schopf, P., et al. (2022) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.2c00574 · Cited by 192
Romero, J., & Aspuru‐Guzik, A. (2021) · Advanced Quantum Technologies · doi:10.1002/qute.202000003 · Cited by 186
A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons
Stair, N. H., Huang, R., & Evangelista, F. A. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.9b01125 · Cited by 183
Strategies for solving the Fermi-Hubbard model on near-term quantum computers
Cade, C., Mineh, L., Montanaro, A., & Stanisic, S. (2020) · Physical Review B · doi:10.1103/physrevb.102.235122 · Cited by 178
“Prior work on variational methods for solving the Hubbard model [17, 19–21] (discussed in Section I) has left a number of important questions open which must be answered to understand whether it is a realistic target for near-term quantum computers.”
BackgroundMethodInfluentialTheory of overparametrization in quantum neural networks
Larocca, M., Ju, N., García-Martín, D., Coles, P. J., & Cerezo, M. (2023) · Nature Computational Science · doi:10.1038/s43588-023-00467-6 · Cited by 262
“Let us first describe the so-called Hamiltonian variational ansatz (HVA) [50, 70].”
BackgroundMethodInfluentialEfficient quantum measurement of Pauli operators in the presence of finite sampling error
Crawford, O., Straaten, B. V., Wang, D., Parks, T., Campbell, E., & Brierley, S. (2021) · Quantum · doi:10.22331/q-2021-01-20-385 · Cited by 188
“[8, 24, 25].”
Low rank representations for quantum simulation of electronic structure
Motta, M., Ye, E., McClean, J. R., Li, Z., Minnich, A. J., Babbush, R., & Chan, G. K. (2021) · npj Quantum Information · doi:10.1038/s41534-021-00416-z · Cited by 207
Emerging quantum computing algorithms for quantum chemistry
Motta, M., & Rice, J. E. (2021) · Wiley Interdisciplinary Reviews Computational Molecular Science · doi:10.1002/wcms.1580 · Cited by 201
“…reducing the measurement cost [347], for example by simultaneously measuring commuting subsets of the Pauli operators needed for the cost function [348–352], leveraging amplitude amplification [353], as well as adopting machine-learning techniques to extract more information from a given…”
MethodTapering off qubits to simulate fermionic HamiltoniansPreprint
Bravyi, S., Gambetta, J., Mezzacapo, A., & Temme, K. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1701.08213 · Cited by 356
Izmaylov, A. F., Yen, T.-C., Lang, R. A., & Verteletskyi, V. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.9b00791 · Cited by 171
Iterative Qubit Coupled Cluster Approach with Efficient Screening of Generators
Ryabinkin, I. G., Lang, R. A., Genin, S. N., & Izmaylov, A. F. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.9b01084 · Cited by 170
Kivlichan, I. D., Gidney, C., Berry, D. W., Wiebe, N., McClean, J., Sun, W., Jiang, Z., Rubin, N., Fowler, A., Aspuru-Guzik, A., et al. (2020) · Quantum · doi:10.22331/q-2020-07-16-296 · Cited by 173
“Low-order Trotter decompositions may be useful in variational algorithms to approximate the ground state energy [28, 77].”
BackgroundVariational algorithms for linear algebra
Xu, X., Sun, J., Endo, S., Li, Y., Benjamin, S. C., & Yuan, X. (2021) · 中国科学通报：英文版 · doi:10.1016/j.scib.2021.06.023 · Cited by 195
“[67, 68] for variational quantum simulation.”
Entanglement devised barren plateau mitigation
Patti, T. L., Najafi, K., Gao, X., & Yelin, S. F. (2021) · Physical Review Research · doi:10.1103/physrevresearch.3.033090 · Cited by 164
“The rapid development of noisy quantum devices [1] has lead to great interest in hybrid quantum-classical variational algorithms, through which classical machine learning techniques are employed to prepare, sample, and optimize states on noisy quantum hardware [2–6].”
BackgroundVariational quantum eigensolver with fewer qubits
Liu, J.-G., Zhang, Y.-H., Wan, Y., & Wang, L. (2019) · Physical Review Research · doi:10.1103/physrevresearch.1.023025 · Cited by 166
Application of fermionic marginal constraints to hybrid quantum algorithms
Rubin, N. C., Babbush, R., & McClean, J. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aab919 · Cited by 152
A non-orthogonal variational quantum eigensolver
Huggins, W. J., Lee, J., Baek, U., O’Gorman, B., & Whaley, K. B. (2020) · New Journal of Physics · doi:10.1088/1367-2630/ab867b · Cited by 148
“While it is believed that even a noisy, modestly-sized quantum computer can prepare and measure states that are out of reach for a classical computer [5], it will still likely be difficult to take advantage of this fact to surpass the capabilities of classical variational methods [6–9].”
BackgroundGroup-Invariant Quantum Machine Learning
Larocca, M., Sauvage, F., Sbahi, F. M., Verdon, G., Coles, P. J., & Cerezo, M. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.030341 · Cited by 180
Fermionic Partial Tomography via Classical Shadows
Zhao, A., Rubin, N. C., & Miyake, A. (2021) · Physical Review Letters · doi:10.1103/physrevlett.127.110504 · Cited by 169
“Initial bounds on the repetition count associated with fermionic two-body Hamiltonians were prohibitively high [9], spurring on much recent work addressing this problem.”
BackgroundInfluentialExponential Error Suppression for Near-Term Quantum Devices
Koczor, B. (2021) · Physical Review X · doi:10.1103/physrevx.11.031057 · Cited by 147
“We prepare the ground state via the usual variational Hamiltonian ansatz (VHA) [11–13], which was proposed in the context of QAOA [22, 75, 76], but has successfully been extended to and analysed in the context of, e.g., quantum chemistry, the Hubbard model as well as spin systems [65, 77–79].”
MethodMethodology for replacing indirect measurements with direct measurements
Mitarai, K., & Fujii, K. (2019) · Physical Review Research · doi:10.1103/physrevresearch.1.013006 · Cited by 139
Variational quantum state eigensolver
Cerezo, M., Sharma, K., Arrasmith, A., & Coles, P. J. (2022) · npj Quantum Information · doi:10.1038/s41534-022-00611-6 · Cited by 174
“VQE promises to revolutionize the field of quantum chemistry [28, 29], and perhaps even nuclear [30] and condensed matter [31, 32] physics.”
BackgroundEfficient variational simulation of non-trivial quantum states
Ho, W. W., & Hsieh, T. H. (2019) · SciPost Physics · doi:10.21468/scipostphys.6.3.029 · Cited by 139
“optimized; the resulting quantum state is then the near to the target state. Such variational quantum approaches have been developed and utilized in a number of contexts, such as in quantum chemistry [24,25], and also in classical optimization problems (for example, as the ‘Quantum Approximate Optimization Algorithm’…”
BackgroundMethodVariational Thermal Quantum Simulation via Thermofield Double States
Wu, J., & Hsieh, T. H. (2019) · Physical Review Letters · doi:10.1103/physrevlett.123.220502 · Cited by 163
Variational Spin-Squeezing Algorithms on Programmable Quantum Sensors
Kaubruegger, R., Silvi, P., Kokail, C., van Bijnen, R., Rey, A. M., Ye, J., Kaufman, A. M., & Zoller, P. (2019) · Physical Review Letters · doi:10.1103/physrevlett.123.260505 · Cited by 133
SU(2) hadrons on a quantum computer via a variational approach
Atas, Y. Y., Zhang, J., Lewis, R., Jahanpour, A., Haase, J. F., & Muschik, C. A. (2021) · Nature Communications · doi:10.1038/s41467-021-26825-4 · Cited by 133
Yao.jl: Extensible, Efficient Framework for Quantum Algorithm Design
Luo, X.-Z., Liu, J.-G., Zhang, P., & Wang, L. (2020) · Quantum · doi:10.22331/q-2020-10-11-341 · Cited by 157
“The identity matrix IMatrix and general permutation matrix PermMatrix [105] are defined in LuxurySparse.jl [65].”
InfluentialBenchmarking Quantum Computers and the Impact of Quantum Noise
Resch, S., & Karpuzcu, U. R. (2021) · ACM Computing Surveys · doi:10.1145/3464420 · Cited by 167
“Currently, classical-quantum hybrid algorithms [69, 70, 71, 72] are popular due to their ability to make use of the limited resources of modern quantum”
BackgroundA non-orthogonal variational quantum eigensolver
Huggins, W., Lee, J., Baek, U., O'Gorman, B., & Whaley, K. (2020) · eScholarship (California Digital Library) · Cited by 128
Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows
Hadfield, C. F., Bravyi, S. B., Raymond, R., & Mezzacapo, A. (2022) · Communications in Mathematical Physics · doi:10.1007/s00220-022-04343-8 · Cited by 135
“variances obtained with our LBCS estimator against other previously known observable estimators that do not increase circuit depth: An estimator based on ‘1 sampling of the Hamiltonian, detailed in [24, 25]. An estimator which measures together collections of qubit-wise commuting Pauli operators. To nd the collections…”
MethodOn barren plateaus and cost function locality in variational quantum algorithms
Uvarov, A. V., & Biamonte, J. D. (2021) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8121/abfac7 · Cited by 126
Quantum Computing and Simulations for Energy Applications: Review and Perspective
Paudel, H. P., Syamlal, M., Crawford, S. E., Lee, Y., Shugayev, R., Lu, P., Ohodnicki, P. R., Mollot, D., & Duan, Y. (2022) · ACS Engineering Au · doi:10.1021/acsengineeringau.1c00033 · Cited by 153
An Adaptive Optimizer for Measurement-Frugal Variational Algorithms
Kübler, J. M., Arrasmith, A., Cincio, L., & Coles, P. J. (2020) · Quantum · doi:10.22331/q-2020-05-11-263 · Cited by 141
“A concern about VHQCAs is that they might require prohibitively many quantum measurements (shots) in order to achieve convergence of the cost function [25], especially for applications like quantum chemistry that require chemical accuracy [26, 27].”
BackgroundHarrow, A. W., & Napp, J. C. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.140502 · Cited by 121
“aint θi= θj There are cases in which one may want to impose a constraint that some parameters are always equal. For example, this situation occurs for the “Hamiltonian variational” ansatz proposed in [WHT15]. Hence, one could have a p-pulse ansatz but a smaller number of independent variationalparameters. We notethat…”
BackgroundFiltering variational quantum algorithms for combinatorial optimization
Amaro, D., Modica, C., Rosenkranz, M., Fiorentini, M., Benedetti, M., & Lubasch, M. (2021) · Quantum Science and Technology · doi:10.1088/2058-9565/ac3e54 · Cited by 144
“F-VQE might also benefit from the original QAOA ansatz [12] or generalizations thereof such as the Hamiltonian variational ansatz [47, 48], the quantum alternating operator ansatz [49], the hardware-efficient mixer-phaser ansatz [50] or the depth optimized QAOA ansatz [51].”
MethodChemistry beyond the scale of exact diagonalization on a quantum-centric supercomputer
Moreno, J. R., Motta, M., Haas, H., Javadi-Abhari, A., Jurcevic, P., Kirby, W., Martiel, S., Sharma, K., Sharma, S., Shirakawa, T., et al. (2025) · Science Advances · doi:10.1126/sciadv.adu9991 · Cited by 265
“Although, in principle, other estimators, such as the standard ones used in variational quantum eigensolvers, have bounded variance for any wave function, the dire scaling of a number of measurements to estimate energies makes them impractical for the molecules targeted in this work ( 48 ).”
Low-depth circuit ansatz for preparing correlated fermionic states on a quantum computer
Dallaire-Demers, P.-L., Romero, J., Veis, L., Sim, S., & Aspuru-Guzik, A. (2019) · Quantum Science and Technology · doi:10.1088/2058-9565/ab3951 · Cited by 119
Measurement reduction in variational quantum algorithms
Zhao, A., Tranter, A., Kirby, W. M., Ung, S. F., Miyake, A., & Love, P. J. (2020) · Physical Review A · doi:10.1103/physreva.101.062322 · Cited by 121
Smart, S. E., & Mazziotti, D. A. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.070504 · Cited by 121
Metadata
- Authors
- Wecker, D., Hastings, M. B., & Troyer, M.
- Affiliations
- Microsoft (United States) (US)
- ETH Zurich (CH)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2015-10-02
- Volume
- 92
- Issue
- 4
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2254754114
- Cited by
- 716
- References
- 15
- Citations by year
- 2026: 48 · 2025: 108 · 2024: 101 · 2023: 97 · 2022: 100 · 2021: 110 · 2020: 69 · 2019: 40 · 2018: 21 · 2017: 11 · 2016: 11
- FWCI
- 14.03
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Microsoft Research
- National Science Foundation (PHY-1066293)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Keywords
- quantum state preparation, adiabatic state preparation, variational ansatz, Hubbard model, unitary coupled cluster, quantum chemistry, circuit depth, NISQ devices
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