Simulated annealing method in variational quantum algorithms
D.A. Aleshin, D.O. Golov, J.V. Tchemarina, A.N. Tsirulev
Tver State University
DOI: 10.26456/pcascnn/2025.17.240
Short communication
Abstract: Variational quantum algorithms are the only quantum algorithms that are currently used in hybrid quantum-classical devices to solve rather practical than modelling problems in condensed matter physics, quantum chemistry, and machine learning. In this paper, a new variational quantum algorithm is analysed in detail, implemented in codes and tested. Its distinctive feature is the use of the simulated annealing algorithm to minimize the objective function (energy) as well as a new type of unitary ansatz with multi-qubit interaction. The quantum part of the algorithm is emulated on a classical computer, and the efficiency of the algorithm is estimated by the number of iterations. To test the algorithm, we choose the problem of finding the ground state of the electronic structure of the hydrogen molecule at the equilibrium distance between protons. The Hamiltonian in the Born-Oppenheimer approximation is modeled by the Hamiltonian of a 4-qubit system in the Pauli basis. The universal ansatz for the problem is constructed taking into account the symmetry of the Hamiltonian as a composition of the exponentials of Pauli operators. It depends on 4 parameters, and the corresponding energy has twelve local minima. The algorithm, implemented as a modular program in Python using both the direct annealing method and the dual_annealing function of the Sci-py library, showed high efficiency in comparison with the algorithm based on the standard ansatz and the gradient descent method.
Keywords: variational quantum algorithm, simulated annealing algorithm, optimization, unitary anzats, Pauli basis, Hamiltonian, ground state
- Dmitry A. Aleshin – 2nd year graduate student, General Mathematics and Mathematical Physics Department, Tver State University
- Dmitriy O. Golov – 2nd year postgraduate student, General Mathematics and Mathematical Physics Department, Tver State University
- Julia V. Tchemarina – Dr. Sc., Dean of Faculty of Mathematics, Tver State University
- Alexander N. Tsirulev – Dr. Sc., Professor, Department of General Mathematics and Mathematical Physics, Tver State University
For citation:
Aleshin D.A., Golov D.O., Tchemarina J.V., Tsirulev A.N. Metod imitatsii otzhiga v variatsionnykh kvantovykh algoritmakh [Simulated annealing method in variational quantum algorithms], Fiziko-khimicheskie aspekty izucheniya klasterov, nanostruktur i nanomaterialov [Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials], 2025, issue 17, pp. 240-249. DOI: 10.26456/pcascnn/2025.17.240. ⎘
Full article (in Russian): download PDF file
References:
1. Peruzzo A., McClean J., Shadbolt P. et al. A variational eigenvalue solver on a photonic quantum processor, Nature Communications, 2014, vol. 5, art. no. 4213, 10 p. DOI: 10.1038/ncomms5213.
2. Tilly J., Chen H, Cao S. et al. The variational quantum eigensolver: a review of methods and best practices, Physics Reports, 2022, vol. 986, pp. 1-128. DOI: 10.1016/j.physrep.2022.08.003.
3. Ryabinkin I.G., Lang R.A., Genin S.N. et al. Qubit coupled cluster method: a systematic approach to quantum chemistry on a quantum computer, Journal of Chemical Theory and Computation, 2018, vol. 14, issue 12, pp. 6317-6326. DOI: 10.1021/acs.jctc.8b00932.
4. McClean J.R., Romero J., Babbush R., Aspuru-Guzik A. The theory of variational hybrid quantum-classical algorithms, New Journal of Physics, 2016, vol. 18, art. no 023023, 23 p. DOI: 10.1088/1367-2630/18/2/023023.
5. Kitaev A.Y., Shen A.H., M. Vyalyi M.N. Classical and quantum computation, Graduate studies in mathematics, vol. 47, trans. L.J. Scnechai. Rhode Island, American Mathematical Society, 2002, 257 p.
6. Chitambar E., Gour G. Quantum resource theories, Review of Modern Physics, 2019, vol. 91, issue 2, art. no 025001, 48 p. DOI: 10.1103/RevModPhys.91.025001.
7. Ingberg L. Simulated Annealing: Practice versus Theory, Mathematical and Computer Modelling, 1993, vol. 18, no 11, pp. 29-57.
8. Salamon P., Sibani P., Frost R. Facts, Conjectures, and Improvements for Simulated Annealing, SIAM Monographs on Mathematical Modeling and Computation, Series no. 7. Philadelphia, Society for Industrial and Applied Mathematics, 2002, 165 p.
9. Lopatin A.A. Metod otzhiga [The annealing method]. Saint Petersburg, Saint Petersburg State University Publ., 2005, 57 p. (In Russian). Available at: www.url: https://math.spbu.ru/user/gran/sb1/lopatin.pdf (accessed 01.07.2025).
10. Golov D.O., Petrov N.A., Tsirulev A.N. Variatsionnyj kvantovyj algoritm dlya malorazmernykh sistem v bazise Pauli [Variational quantum algorithm for low-dimensional systems in the Pauli basis], Fiziko-khimicheskie aspekty izucheniya klasterov, nanostruktur i nanomaterialov [Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials], 2024, issue 16, pp. 343-350. DOI: 10.26456/pcascnn/2024.16.343 (In Russian).
11. Szabo A., Ostlund N.S. Modern quantum chemistry: Introduction to advanced electronic structure theory. New York, Dover Publication, 481 p.
12. Du Y., Huang T., You S. et al. Quantum circuit architecture search for variational quantum algorithms, npj Quantum Information, 2022, vol. 8, art. no 62, 8 p. DOI: 10.1038/s41534-022-00570-y.
13. Du Y., Huang T., You S. et al. Supplementary information for: «Quantum circuit architecture search for variational quantum algorithms», npj Quantum Information, 2022, vol. 8, art. no 62, 14 p. Available at:: www.url:
https://static-content.springer.com/esm/art%3A10.1038%2Fs41534-022-00570 y/MediaObjects/41534_2022_570_MOESM1_ESM.pdf (accessed 01.07.2025).
14. Tsirulev A.N. A geometric view on quantum tensor networks, European Physical Journal, 2020, vol. 226, issue 4, art. no. 02022, 5 p. DOI: 10.1051/epjconf/202022602022.
15. Nikonov V.V., Tsirulev A.N. Pauli basis formalism in quantum computations, Mathematical Modelling and Geometry, 2020, vol. 8, no 3, pp. 1–14. DOI: 10.26456/mmg/2020-831.
16. Taube A.G, Bartlett R.J. New perspectives on unitary coupled-cluster theory, International Journal of Quantum Chemistry, 2006, vol. 106, issue 15, pp. 3393-3401. DOI: 10.1002/qua.21198.
17. Andre E., Tsirulev A.N. Modelirovanie zaputannykh sostoyanij v klasterakh kubitov [Modeling of entangled states in qubit clusters], Fiziko-khimicheskie aspekty izucheniya klasterov, nanostruktur i nanomaterialov [Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials], 2022, issue 14, pp. 143-146. DOI: 10.26456/pcascnn/2022.14.342. (In Russian).
18. Annealing Method Python. Available at: www.url: https://github.com/KoTuK2306/masters_thesis/tree/master/AnnealingMethodPython (accessed 01.07.2025).
19. R3DDG / Variational Quantum Eigensolver - with - Annealing - optimization. Available at: www.url: https://github.com/R3DDG/VariationalQuantumEigensolver-with-Annealing-optimization (accessed 01.07.2025).