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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]
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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 McClean et al., 2016
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“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
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“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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“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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“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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Citations
Cites
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Self-citationThe Mathematical Theory of Optimal Processes
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Chukwu, U., Dridi, R., Berwald, J., Booth, M., Dawson, J., Le, D., Wainger, M., & Reinhardt, S. P. (2020) · 2020 IEEE High Performance Extreme Computing Conference (HPEC) · doi:10.1109/hpec43674.2020.9286230 · Cited by 9
Input Redundancy for Parameterized Quantum Circuits
Vidal, J. G., & Theis, D. O. (2019) · arXiv · doi:10.48550/arxiv.1901.11434 · Cited by 6
Quinones, M. P., & Junqueira, C. (2020) · arXiv · doi:10.48550/arxiv.2007.13245 · Cited by 6
Limitations of Quantum Hardware for Molecular Energy Estimation Using VQE
Carreras, A., Orús, R., & Casanova, D. (2026) · Physical Chemistry Chemical Physics · doi:10.1039/d5cp03907j · Cited by 5
Implementing a Variational Quantum Eigensolver using Superconducting Qubits
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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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