Quantum optimization using variational algorithms on near-term quantum devices
Open accessAbstract
Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed. With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupled-cluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.
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Existing quantum hardware is not suited to implement deep and highly connected circuits required for the UCC and similar ansätze for applications beyond basic demonstrations such as the H2 molecule.
“Therefore, existing quantum hardware is not suited to implement deep and highly connected circuits required for the UCC and similar ansätze for applications beyond basic demonstrations such as the H 2 molecule (Moll et al. , 2018).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Moll et al., 2018
The Trotter error of variational fast forwarding can be removed by diagonalizing instead the Hamiltonian H that generates the evolution.
“As shown in Ref. [125], the ensuing Trotter error of this approach can be removed by diagonalizing instead the Hamiltonian H that generates the evolution.”
From Variational quantum algorithms (2021) · cites this paper as [125]
Section 5.1 mentions a trotterized adiabatic algorithm, while section 3.1 describes variational eigenstate preparation; neither addresses removing Trotter error by diagonalizing the evolution Hamiltonian. Searches for “variational fast forwarding,” “Trotter error,” and Hamiltonian diagonalization found no passage supporting the attributed claim. Page numbers were not available in our copy. (LLM, conf=0.77)
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Reiner, J.-M., Marthaler, M., Braumüller, J., Weides, M. P., & Schön, G. (2016) · Physical Review A · doi:10.1103/physreva.94.032338 · Cited by 40 · see citing works
Quantum Algorithms for Quantum Chemistry based on the sparsity of the CI-matrixPreprint
Toloui, B., & Love, P. J. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1312.2579 · Cited by 40 · see citing works
From local to global ground states in Ising spin glasses
Zintchenko, I., Hastings, M. B., & Troyer, M. (2015) · Physical Review B · doi:10.1103/physrevb.91.024201 · Cited by 31 · see citing works
Mueck, L. (2015) · Nature Chemistry · doi:10.1038/nchem.2248 · Cited by 24 · see citing works
Quantum chemistry and charge transport in biomolecules with superconducting circuits
García-Álvarez, L., Heras, U. L., Mezzacapo, A., Sanz, M., Solano, E., & Lamata, L. (2016) · Scientific Reports · doi:10.1038/srep27836 · Cited by 23 · see citing works
Self-citationTempel, D. G., & Aspuru‐Guzik, A. (2012) · Scientific Reports · doi:10.1038/srep00391 · Cited by 20 · see citing works
Unified views of quantum simulation algorithms for chemistryPreprint
Whitfield, J. D. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1502.03771 · Cited by 1 · see citing works
Flagged first, then by citations
Research works printed but not linked: 6
- Albash T and Lidar D A 2017 Evidence for a limited quantum speedup on a quantum annealer arXiv:1705.07452 [quant-ph] · arXiv
- Jones N C, Whitfield J D, McMahon P L, Yung M H, Meter R V, Aspuru-Guzik A and Yamamoto Y 2012 Faster quantum chemistry simulation on fault-tolerant quantum computers New J. Phys. 14 115023 · DOI · Paper page
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Cited by
Sorted by citations
Variational quantum algorithms
Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S. C., Endo, S., Fujii, K., McClean, J. R., Mitarai, K., Yuan, X., Cincio, L., et al. (2021) · Nature Reviews Physics · doi:10.1038/s42254-021-00348-9 · Cited by 3,385 · see citing works
“Here we discuss a different possibility where one employs a VQA to solve a classical optimization problem [105].”
BackgroundBarren plateaus in quantum neural network training landscapes
McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R., & Neven, H. (2018) · Nature Communications · doi:10.1038/s41467-018-07090-4 · Cited by 2,329 · see citing works
Noisy intermediate-scale quantum algorithms
Bharti, K., Cervera-Lierta, A., Kyaw, T. H., Haug, T., Alperin-Lea, S., Anand, A., Degroote, M., Heimonen, H., Kottmann, J. S., Menke, T., et al. (2022) · Reviews of Modern Physics · doi:10.1103/revmodphys.94.015004 · Cited by 1,726 · see citing works
Superconducting Qubits: Current State of Play
Kjaergaard, M., Schwartz, M. E., Braumüller, J., Krantz, P., Wang, J. I.-J., Gustavsson, S., & Oliver, W. D. (2020) · Annual Review of Condensed Matter Physics · doi:10.1146/annurev-conmatphys-031119-050605 · Cited by 1,544 · see citing works
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 · see citing works
Quantum Algorithms for Quantum Chemistry and Quantum Materials Science
Bauer, B., Bravyi, S., Motta, M., & Chan, G. K.-L. (2020) · Chemical Reviews · doi:10.1021/acs.chemrev.9b00829 · Cited by 848 · see citing works
Motta, M., Sun, C., Tan, A. T. K., O’Rourke, M. J., Ye, E., Minnich, A. J., Brandão, F. G. S. L., & Chan, G. K.-L. (2020) · Nature Physics · doi:10.1038/s41567-019-0704-4 · Cited by 709 · see citing works
Validating quantum computers using randomized model circuits
Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D., & Gambetta, J. M. (2019) · Physical Review A · doi:10.1103/physreva.100.032328 · Cited by 704 · see citing works
Self-citationZhou, L., Wang, S.-T., Choi, S., Pichler, H., & Lukin, M. D. (2020) · Physical Review X · doi:10.1103/physrevx.10.021067 · Cited by 913 · see citing works
Self-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 · see citing works
Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation
Endo, S., Cai, Z., Benjamin, S. C., & Yuan, X. (2021) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.90.032001 · Cited by 566 · see citing works
Variational Quantum Computation of Excited States
Higgott, O., Wang, D., & Brierley, S. (2019) · Quantum · doi:10.22331/q-2019-07-01-156 · Cited by 463 · see citing works
Continuous-variable quantum neural networks
Killoran, N., Bromley, T. R., Arrazola, J. M., Schuld, M., Quesada, N., & Lloyd, S. (2019) · Physical Review Research · doi:10.1103/physrevresearch.1.033063 · Cited by 418 · see citing works
A generative modeling approach for benchmarking and training shallow quantum circuits
Benedetti, M., Garcia-Pintos, D., Perdomo, O., Leyton-Ortega, V., Nam, Y., & Perdomo-Ortiz, A. (2019) · npj Quantum Information · doi:10.1038/s41534-019-0157-8 · Cited by 379 · see citing works
Barkoutsos, P. K., Gonthier, J. F., Sokolov, I., Moll, N., Salis, G., Fuhrer, A., Ganzhorn, M., Egger, D. J., Troyer, M., Mezzacapo, A., et al. (2018) · Physical Review A · doi:10.1103/physreva.98.022322 · Cited by 363 · see citing works
Self-citationWarm-starting quantum optimization
Egger, D. J., Mareček, J., & Woerner, S. (2021) · Quantum · doi:10.22331/q-2021-06-17-479 · Cited by 269 · see citing works
Self-citationLow-cost error mitigation by symmetry verification
Bonet-Monroig, X., Sagastizabal, R., Singh, M., & O'Brien, T. E. (2018) · Physical Review A · doi:10.1103/physreva.98.062339 · Cited by 250 · see citing works
Improving Variational Quantum Optimization using CVaR
Barkoutsos, P. K., Nannicini, G., Robert, A., Tavernelli, I., & Woerner, S. (2020) · Quantum · doi:10.22331/q-2020-04-20-256 · Cited by 263 · see citing works
Self-citationHarnessing the Power of the Second Quantum Revolution
Deutsch, I. H. (2020) · PRX Quantum · doi:10.1103/prxquantum.1.020101 · Cited by 207 · 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
Sokolov, I. O., Barkoutsos, P. K., Ollitrault, P. J., Greenberg, D., Rice, J., Pistoia, M., & Tavernelli, I. (2020) · The Journal of Chemical Physics · doi:10.1063/1.5141835 · Cited by 164 · see citing works
Self-citationQuantum many-body simulations on digital quantum computers: State-of-the-art and future challenges
Fauseweh, B. (2024) · Nature Communications · doi:10.1038/s41467-024-46402-9 · Cited by 152 · see citing works
Performance of hybrid quantum-classical variational heuristics for combinatorial optimization
Nannicini, G. (2019) · Physical Review E · doi:10.1103/physreve.99.013304 · Cited by 138 · 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
Variational Quantum Classifier for Binary Classification: Real vs Synthetic Dataset
Maheshwari, D., Sierra-Sosa, D., & Garcia-Zapirain, B. (2022) · IEEE Access · doi:10.1109/access.2021.3139323 · Cited by 108 · see citing works
$O(N^3)$ Measurement Cost for Variational Quantum Eigensolver on Molecular Hamiltonians
Gokhale, P., Angiuli, O., Ding, Y., Gui, K., Tomesh, T., Suchara, M., Martonosi, M., & Chong, F. T. (2020) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2020.3035814 · Cited by 104 · see citing works
Universal variational quantum computation
Biamonte, J. (2021) · Physical Review A · doi:10.1103/physreva.103.l030401 · Cited by 102 · see citing works
Variational-state quantum metrology
Koczor, B., Endo, S., Jones, T., Matsuzaki, Y., & Benjamin, S. C. (2020) · New Journal of Physics · doi:10.1088/1367-2630/ab965e · Cited by 101 · see citing works
Layer VQE: A Variational Approach for Combinatorial Optimization on Noisy Quantum Computers
Liu, X., Angone, A., Shaydulin, R., Safro, I., Alexeev, Y., & Cincio, L. (2022) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2021.3140190 · Cited by 78 · see citing works
Optimizing quantum heuristics with meta-learning
Wilson, M., Stromswold, R., Wudarski, F., Hadfield, S., Tubman, N. M., & Rieffel, E. G. (2021) · Quantum Machine Intelligence · doi:10.1007/s42484-020-00022-w · Cited by 79 · see citing works
Classification with Quantum Machine Learning: A Survey
Abohashima, Z., Elhosen, M., Houssein, E. H., & Mohamed, W. M. (2020) · arXiv · doi:10.48550/arxiv.2006.12270 · Cited by 68 · see citing works
Quantum singular value decomposer
Bravo-Prieto, C., García-Martín, D., & Latorre, J. I. (2020) · Physical Review A · doi:10.1103/physreva.101.062310 · Cited by 64 · see citing works
Variational learning of Grover's quantum search algorithm
Morales, M. E. S., Tlyachev, T., & Biamonte, J. (2018) · Physical Review A · doi:10.1103/physreva.98.062333 · Cited by 57 · see citing works
Efficient Encoding of the Weighted MAX $$k$$-CUT on a Quantum Computer Using QAOA
Fuchs, F. G., Kolden, H. Ø., Aase, N. H., & Sartor, G. (2021) · SN Computer Science · doi:10.1007/s42979-020-00437-z · Cited by 56 · see citing works
Multiblock ADMM Heuristics for Mixed-Binary Optimization on Classical and Quantum Computers
Gambella, C., & Simonetto, A. (2020) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2020.3033139 · Cited by 60 · see citing works
Baldwin, C. H., Mayer, K., Brown, N. C., Ryan-Anderson, C., & Hayes, D. (2022) · Quantum · doi:10.22331/q-2022-05-09-707 · Cited by 54 · see citing works
Quantum variational optimization: The role of entanglement and problem hardness
Díez-Valle, P., Porras, D., & García-Ripoll, J. J. (2021) · Physical Review A · doi:10.1103/physreva.104.062426 · Cited by 47 · see citing works
“In quantum variational algorithms [2, 35], the quantum computer is used to prepare and characterize a complex wavefunction, defined in terms of parameterized quantum operations”
MethodAbrupt transitions in variational quantum circuit training
Campos, E., Nasrallah, A., & Biamonte, J. (2021) · Physical Review A · doi:10.1103/physreva.103.032607 · Cited by 46 · see citing works
Nonunitary Operations for Ground-State Calculations in Near-Term Quantum Computers
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Self-citationCastaldo, D., Rosa, M., & Corni, S. (2020) · arXiv · doi:10.48550/arxiv.2007.00368 · Cited by 32
Max-independent set and the quantum alternating operator ansatz
Saleem, Z. H. (2020) · International Journal of Quantum Information · doi:10.1142/s0219749920500112 · Cited by 28 · see citing works
Quinones, M. P., & Junqueira, C. (2020) · arXiv · doi:10.48550/arxiv.2007.13245 · Cited by 6 · see citing works
- Authors
- Moll, N., Barkoutsos, P., Bishop, L. S., Chow, J. M., Cross, A., Egger, D. J., Filipp, S., Fuhrer, A., Gambetta, J. M., Ganzhorn, M., Kandala, A., Mezzacapo, A., Müller, P., Riess, W., Salis, G., Smolin, J., Tavernelli, I., & Temme, K.
- Affiliations
- IBM Research - Zurich (CH)
- IBM Research - Thomas J. Watson Research Center (US)
- Corresponding author
- Moll, Nikolaj
- ORCID
- Venue
- Quantum Science and Technology
- Publisher
- IOP Publishing
- Published
- 2018-06-19
- Volume
- 3
- Issue
- 3
- Pages
- 030503
- ISSN
- 2058-9565 (ISSN-L)
- OpenAlex
- W2760965790
- Cited by
- 744
- References
- 104
- Citations by year
- 2026: 56 · 2025: 103 · 2024: 96 · 2023: 133 · 2022: 109 · 2021: 111 · 2020: 81 · 2019: 42 · 2018: 11 · 2017: 2
- FWCI
- 34.02
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Quantum Science and Technology (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- 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 › Advanced Chemical Physics Studies
- Keywords
- variational quantum algorithms, quantum optimization, near-term quantum devices, hybrid quantum-classical algorithms, quantum chemistry, quantum volume, fermion-to-qubit mapping, molecular ground state, error mitigation, ground state energy estimation, entanglement
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