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The theory of variational hybrid quantum-classical algorithms
Open accessAbstract
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 (Peruzzo et al 2014 Nat. Commun. 5 4213 ) 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. Additionally, 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 derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.
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“To this end, it has been shown that by considering general commutations (and increasing the number of gates of the circuit quadratically with n ) the scaling of the number of measurements can be reduced to n 3 [213, 214, 216-219].”
From Variational quantum algorithms (2021) · cites this paper as [218]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“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 McClean et al., 2016
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“The electronic structure problem is one of the most prominent tasks within VQAs (see for example the reviews (Cao et al. , 2019; McArdle et al. , 2020)) and was the pioneering task for VQE (McClean et al. , 2016; Peruzzo et al. , 2014).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as McClean et al. , 2016
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“An important class of algorithms for NISQ hardware are socalled hybrid quantum-classical algorithms that partition the computational workload between a classical and a quantum computer.[124,125]”
From Quantum Computing for Molecular Biology** (2023) · cites this paper as [124]
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In the citing paper
“Pauli strings that commute qubit-wise, i.e. the Pauli operators on each qubit commute, can be measured at the same time (Kandala et al. , 2017; McClean et al. , 2016).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as McClean et al., 2016
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In the citing paper
“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 McClean et al. , 2016
In the citing paper
“In contrast, the Boson Sampling model of computation does not have this property 63 .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [63]
In the citing paper
“Many algorithms for near-term quantum computers are based on variational circuits [1-6]”
From Estimating the gradient and higher-order derivatives on quantum hardware (2021) · cites this paper as [4]
In the citing paper
“This definition differs from [13, 14] and loosely coincides with the complexity of classically simulating the model circuits.”
From Validating quantum computers using randomized model circuits (2019) · cites this paper as [14]
In the citing paper
“Hybrid quantum-classical algorithms, such as the variational quantum eigensolver 2,3 (VQE) and the quantum approximate optimization algorithm 4,5 (QAOA), provide ways to use NISQ computers for practical applications.”
From A generative modeling approach for benchmarking and training shallow quantum circuits (2019) · cites this paper as [3]
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“Furthermore, in our proposal, the preparation of the ground state could be practically realized through a variational quantum eigensolver [103, 104], with additional variational parameters related to the local basis optimizations.”
Hong, C.-L., Tsai, T., Chou, J., Chen, P.-J., Tsai, P.-K., Chen, Y.-C., Kuo, E.-J., Srolovitz, D. J., Hu, A., Cheng, Y., et al. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.020360 · Cited by 18
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“Two representative HQC algorithms are the variational quantum eigensolver (VQE) (Peruzzo et al., 2013; Wecker et al., 2015; McClean et al., 2016; O’Malley et al., 2016; Santagati et al., 2016; Guerreschi and Smelyanskiy, 2017; McClean et al., 2017; Romero et al., 2017; Shen et al., 2017) and the…”
BackgroundMethodSelf-citationProgress toward practical quantum cryptanalysis by variational quantum cloning
Coyle, B., Doosti, M., Kashefi, E., & Kumar, N. (2022) · Physical Review A · doi:10.1103/physreva.105.042604 · Cited by 19
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Blueprint for a digital-analog variational quantum eigensolver using Rydberg atom arrays
Michel, A., Grijalva, S., Henriet, L., Domain, C., & Browaeys, A. (2023) · Physical Review A · doi:10.1103/physreva.107.042602 · Cited by 14
Efficient quantum simulation of electron-phonon systems by variational basis state encoder
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Sawaya, N. P. D., Schmitz, A. T., & Hadfield, S. (2023) · Quantum · doi:10.22331/q-2023-09-14-1111 · Cited by 28
Restoring symmetries in quantum computing using Classical Shadows
Guzman, E. A. R., & Denis Lacroix, A. (2024) · The European Physical Journal A · doi:10.1140/epja/s10050-024-01314-6 · Cited by 14
“Then the technique can be considered as a practical method to implement the SB-SR approach into the conventional Variational Quantum Eigensolver (VQE) [7,8,24,25].”
Variational quantum algorithms for simulation of Lindblad dynamics
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Rao, G. V. E., & Rajitha, B. (2024) · International Journal of Information Technology · doi:10.1007/s41870-023-01681-1 · Cited by 14
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VarSaw: Application-tailored Measurement Error Mitigation for Variational Quantum Algorithms
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“In this work, we use a variational quantum eigensolver (VQE) [7, 26, 27] to obtain approximate ground states; subsequently used as trial wavefunctions in AFQMC.”
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Secure Delegated Variational Quantum Algorithms
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“They have the potential to demonstrate quantum advantages in the noisy intermediate-scale quantum (NISQ) era [1], [2], [3], as they have shown their intrinsic resilience to noise and even benefited from it [4], making them highly compatible with near-term quantum devices.”
BackgroundSubspace Methods for the Simulation of Molecular Response Properties on a Quantum Computer
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“4,5 The variational quantum eigensolver (VQE) approach 6,7 attempts to circumvent this problem with a hybrid quantum-classical workflow in which quantum and classical processors cooperate.”
MethodQuantum-Social Network Analysis for Community Detection: A Comprehensive Review
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“Quantum simulation is an alternative method for hybrid quantum-classical community detection [22], [23].”
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“These algorithms are known as variational quantum algorithms (VQA) [14–19], with some notable examples including the Quantum Approximate Optimization Algorithm (QAOA) [20–22] and the Variational Quantum Eigensolver (VQE) [23–26].”
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Halder, S., Shrikhande, C., & Maitra, R. (2023) · The Journal of Chemical Physics · doi:10.1063/5.0166433 · Cited by 13
Benchmarking universal quantum gates via channel spectrum
Gu, Y., Zhuang, W.-F., Chai, X., & Liu, D. E. (2023) · Nature Communications · doi:10.1038/s41467-023-41598-8 · Cited by 14
Phenomenological theory of variational quantum ground-state preparation
Astrakhantsev, N., Mazzola, G., Tavernelli, I., & Carleo, G. (2023) · Physical Review Research · doi:10.1103/physrevresearch.5.033225 · Cited by 22
Manenti, R., & Motta, M. (2023) · doi:10.1093/oso/9780198787488.001.0001 · Cited by 15
Deep Neural Network Assisted Quantum Chemistry Calculations on Quantum Computers
Ghosh, K., Kumar, S., Nirmal, M. R., & Yamijala, S. S. R. K. C. (2023) · ACS Omega · doi:10.1021/acsomega.3c07364 · Cited by 13
Tailored and Externally Corrected Coupled Cluster with Quantum Inputs
Scheurer, M., Anselmetti, G.-L., Oumarou, O., Gogolin, C., & Rubin, N. C. (2024) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.4c00037 · Cited by 18
“…chemistry, e.g., virtual quantum subspace expansion (VQSE), 4 second-order perturbation theory using the variational quantum eigensolver (VQE) 1,5,6 together with quantum subspace expansion (QSE), named NEVPT2(VQE,QSE), 7 non-orthogonal configuration interaction approaches, 8 and NEVPT2 based on…”
MethodGrover-QAOA for 3-SAT: quadratic speedup, fair-sampling, and parameter clustering
Zhang, Z., Paredes, R., Sundar, B., Quiroga, D., Kyrillidis, A., Dueñas‐Osorio, L., Pagano, G., & Hazzard, K. R. A. (2024) · Quantum Science and Technology · doi:10.1088/2058-9565/ad895c · Cited by 25
“Combinatorial optimization problems are of great interest for quantum computing and have been explored using amplitude amplification [1], quantum annealing [2], and variational techniques, such as the variational quantum eigensolver [3] and the quantum approximate optimization algorithm (QAOA) [4].”
Programmable adiabatic demagnetization for systems with trivial and topological excitations
Matthies, A., Rudner, M. S., Rosch, A., & Berg, E. (2024) · Quantum · doi:10.22331/q-2024-10-23-1505 · Cited by 25
Solving an industrially relevant quantum chemistry problem on quantum hardware
Nützel, L., Gresch, A., Hehn, L., Marti, L., Freund, R., Steiner, A., Marciniak, C. D., Eckstein, T., Stockinger, N., Wolf, S., et al. (2024) · Quantum Science and Technology · doi:10.1088/2058-9565/ad9ed3 · Cited by 18
Marti-Dafcik, D., Burton, H. G. A., & Tew, D. P. (2025) · Physical Review Research · doi:10.1103/physrevresearch.7.013191 · Cited by 20
“[11] These estimates are then passed to a classical computer, which uses traditional optimization algorithms to proposes updates to θ that minimise E ( θ ).”
Nengroo, S. H., Har, D., Jeong, H., Heo, T., & Lee, S. (2025) · Energy and AI · doi:10.1016/j.egyai.2025.100541 · Cited by 13
VanQver: the variational and adiabatically navigated quantum eigensolver
Matsuura, S., Yamazaki, T., Senicourt, V., Huntington, L., & Zaribafiyan, A. (2020) · New Journal of Physics · doi:10.1088/1367-2630/ab8080 · Cited by 13
Variational quantum state preparation via quantum data buses
Kuzmin, V. V., & Silvi, P. (2020) · Quantum · doi:10.22331/q-2020-07-06-290 · Cited by 12
Lowering tomography costs in quantum simulation with a symmetry projected operator basis
Smart, S. E., & Mazziotti, D. A. (2021) · Physical Review A · doi:10.1103/physreva.103.012420 · Cited by 13
Hybrid quantum-classical algorithms for solving quantum chemistry in Hamiltonian–wave-function space
Yuan, Z.-H., Yin, T., & Zhang, D.-B. (2021) · Physical Review A · doi:10.1103/physreva.103.012413 · Cited by 13
Computing Eigenvalues of Diagonalizable Matrices on a Quantum Computer
Shao, C. (2022) · ACM Transactions on Quantum Computing · doi:10.1145/3527845 · Cited by 16
Numerical hardware-efficient variational quantum simulation of a soliton solution
Kardashin, A., Pervishko, A. A., Biamonte, J., & Yudin, D. I. (2021) · Physical Review A · doi:10.1103/physreva.104.l020402 · Cited by 16
Normalized Gradient Descent for Variational Quantum Algorithms
Suzuki, Y., Yano, H., Raymond, R., & Yamamoto, N. (2021) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce52317.2021.00015 · Cited by 13
Wave Function Adapted Hamiltonians for Quantum Computing
Ratini, L., Capecci, C., Benfenati, F., & Guidoni, L. (2022) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.1c01170 · Cited by 15
Peel | Pile? Cross-Framework Portability of Quantum Software
Schönberger, M., Franz, M., Scherzinger, S., & Mauerer, W. (2022) · IEEE International Conference on Software Architecture Companion (ICSA-C) · doi:10.1109/icsa-c54293.2022.00039 · Cited by 13
Model-Free Deep Recurrent Q-Network Reinforcement Learning for Quantum Circuit Architectures Design
Sogabe, T., Kimura, T., Chen, C.-C., Shiba, K., Kasahara, N., Sogabe, M., & Sakamoto, K. (2022) · Quantum Reports · doi:10.3390/quantum4040027 · Cited by 13
Sevriuk, V., Liu, W., Rönkkö, J., et al. (2022) · Applied Physics Letters · doi:10.1063/5.0129345 · Cited by 13
Variational dynamics as a ground-state problem on a quantum computer
Barison, S., Vicentini, F., Cirac, J. I., & Carleo, G. (2022) · Physical Review Research · doi:10.1103/physrevresearch.4.043161 · Cited by 19
Metadata
- Authors
- McClean, J. R., Romero, J., Babbush, R., & Aspuru-Guzik, A.
- Affiliations
- Lawrence Berkeley National Laboratory (US)
- Harvard University (US)
- Google (United States) (US)
- Corresponding author
- McClean, Jarrod R
- Venue
- New Journal of Physics
- Publisher
- IOP Publishing
- Published
- 2016-02-04
- Volume
- 18
- Issue
- 2
- Pages
- 023023
- ISSN
- 1367-2630 (ISSN-L)
- OpenAlex
- W2257937122
- Semantic Scholar
- c78988d6c8b3d0a0385164b372f202cdeb4a5849
- Cited by
- 2,227
- References
- 93
- Citations by year
- 2026: 219 · 2025: 353 · 2024: 306 · 2023: 321 · 2022: 310 · 2021: 337 · 2020: 203 · 2019: 99 · 2018: 50 · 2017: 18 · 2016: 9
- FWCI
- 48.22
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- New Journal of Physics (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- Digital Access to Scholarship at Harvard (DASH) (Harvard University) (submittedVersion, cc-by)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
- OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Army Research Office (W911NF-15-1-0256)
- Air Force Office of Scientific Research (FA9550-12-1-0046)
- Office of Naval Research Global (N00014-16-1-2008)
- Defense Security Science Engineering Fellowship (N00014-16-1-2008)
- Lawrence Berkeley National Laboratory (Luis W. Alvarez Fellowship in Compu)
- 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
- hybrid quantum-classical algorithms, unitary coupled cluster, universal gate set, truncation, correlated sampling, derivative-free optimization
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