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 quantum machine learning algorithms based on parameterized quantum circuits have demonstrated superior performance and computational abilities compared to conventional algorithms.
“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]
The opening summary (p. 1) reports that a parameterized quantum-circuit ansatz converged faster than earlier unitary-coupled-cluster schemes. The Discussion (p. 10) describes outperforming classical computers as a prospect, not a demonstrated result. The paper therefore does not establish the cited claim that several quantum machine-learning algorithms have demonstrated superior performance over conventional algorithms. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThe Variational Hamiltonian Ansatz was developed to reduce the number of parameters and accelerate convergence.
“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
The Introduction (pp. 1–2) describes a Hamiltonian variational ansatz inspired by adiabatic state preparation, with few parameters and a shorter sequence of rotations. The faster convergence reported in “Optimization with Exact Energies” (p. 4) is specifically for an annealed optimization procedure using that ansatz, not a general property established for the ansatz itself. (LLM, conf=0.82)
Experimental — accuracy not yet measuredVQE has been proposed to find the ground state of a Hamiltonian in a manner suited for NISQ devices.
“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
The AQC model is computationally equivalent to the gate model, meaning algorithms written in either model can be translated to the other without prohibitive computational overhead.
“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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“These advantages are significant given the high accuracy required in quantum chemical applications (24).”
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Wiebe, N., & Granade, C. (2016) · Physical Review Letters · doi:10.1103/physrevlett.117.010503 · Cited by 193 · see citing works
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 195 · see citing works
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 · see citing works
Romero, J., & Aspuru‐Guzik, A. (2021) · Advanced Quantum Technologies · doi:10.1002/qute.202000003 · Cited by 186 · see citing works
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 · see citing works
“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.”
BackgroundMethodInfluentialA 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 · see citing works
Theory 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 191 · see citing works
“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 · see citing works
“[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 179 · see citing works
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 180 · see citing works
“…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. B., Gambetta, J., Mezzacapo, A., & Temme, K. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1701.08213 · Cited by 172 · see citing works
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 · see citing works
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 · see citing works
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 · see citing works
“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 170 · see citing works
“[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 · see citing works
“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 · see citing works
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 · see citing works
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 · see citing works
“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 145 · see citing works
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 142 · see citing works
“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 · see citing works
“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 · see citing works
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 154 · see citing works
“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 · see citing works
“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 137 · see citing works
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 · see citing works
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 · see citing works
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 · see citing works
“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 134 · see citing works
“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 · see citing works
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 · see citing works
“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 · see citing works
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 132 · see citing works
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 · see citing works
“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 · see citing works
“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 123 · see citing works
“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 141 · see citing works
“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 · see citing works
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 · see citing works
Smart, S. E., & Mazziotti, D. A. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.070504 · Cited by 117 · see citing works
Performance comparison of optimization methods on variational quantum algorithms
Bonet-Monroig, X., Wang, H., Vermetten, D., Senjean, B., Moussa, C., Bäck, T., Dunjko, V., & O’Brien, T. E. (2023) · Physical Review A · doi:10.1103/physreva.107.032407 · Cited by 119 · see citing works
Measuring the Rényi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computer
Linke, N. M., Johri, S., Figgatt, C., Landsman, K. A., Matsuura, A. Y., & Monroe, C. R. (2018) · Physical Review A · doi:10.1103/physreva.98.052334 · Cited by 109 · see citing works
- 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 › Computer Science › Artificial Intelligence › 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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