Simulating Ising-Like Models on Quantum Computer

DOI: https://doi.org/10.25559/SITITO.019.202303.554-563

Abstract

Due to the complexity of lattice models of statistical physics, there is interest in developing new approaches to study them, including those using quantum technologies. In this paper, we describe and implement a scheme for applying the Variational Quantum Eigensolver to the problem of finding the free energy and magnetization in the thermodynamic limit of the n-chain generalized planar Ising model with the interaction of nearest neighbors, next nearest neighbors, and plaquette interactions. The calculation of Rayleigh quotient for transfer matrix is described in detail for n = 1,2,3. Using special parameterizations of the state of the system of qubits and transfer matrix decomposition, free energy and magnetization are calculated for 3-chain model on a quantum computer emulator. The entire computation process, including quantum computer emulation, is implemented using the Python programming language. Also, a method is proposed for significant acceleration of calculations (by about 10.000 times) on the emulator for considered models. Confidence intervals are constructed for the found characteristics of the model.

Author Biographies

Andrey Sergeevich Andreev, Bauman Moscow State Technical University

Student of the Department of Higher Mathematics, Faculty of Fundamental Sciences

Pavel Vasilevich Khrapov, Bauman Moscow State Technical University

Associate Professor of the Department of Higher Mathematics, Faculty of Fundamental Sciences, Cand. Sci. (Phys.-Math.)

References

1. Yurishchev . . KteoriidvojnyhcepejIzinga. Model' vo vneshnem pole [To the Theory of Double Ising Chains. A Model in an External Field]. Fizika Nyzkikh Temperatur = Low Temperature Physics. 1979;(5):477-483. (In Russ.)
2. Yurishchev .A. K teorii dvojnyh cepej Izinga. Model' v nulevom vneshnem pole [On the Theory of Double Ising Chains. The Model in Zero External Field]. Fizika Nyzkikh Temperatur = Low Temperature Physics.1978;(4):646-654. (In Russ.).[TA1]
3. Abhijith J., Adedoyin A., Ambrosiano J. et al. Quantum Algorithm Implementations for Beginners. ACM Transactions on Quantum Computing. 2022;3(4):18. https://doi.org/10.1145/3517340
4. Poulin D., Wocjan P. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Physical Review Letters. 2009;103(22):220502.https://doi.org/10.1103/PhysRevLett.103.220502
5. Matsumoto K. et al. Calculation of Gibbs partition function with imaginary time evolution on near-term quantum computers. Japanese Journal of Applied Physics. 2022;61(4):042002. https://doi.org/10.35848/1347-4065/ac5152
6. Geraci J., Lidar D. A. On the exact evaluation of certain instances of the Potts partition function by quantum computers. Communications in Mathematical Physics. 2008;279:735-768. https://doi.org/10.1007/s00220-008-0438-0
7. Fujii K., Morimae T. Commuting quantum circuits and complexity of Ising partition functions. New Journal of Physics. 2017;19(3):033003. https://doi.org/10.1088/1367-2630/aa5fdb
8. Wu Y., Wang J.B. Estimating Gibbs partition function with quantum Clifford sampling. Quantum Science and Technology. 2022;7(2):025006. https://doi.org/10.1088/2058-9565/ac47f0
9. Jackson A., Kapourniotis T., Datta A. Partition-function estimation: Quantum and quantum-inspired algorithms. Physical Review A. 2023;107(1):012421. https://doi.org/10.1103/PhysRevA.107.012421
10. van Dijk J., Prodan E. Vertex Lattice Models Simulated with Quantum Circuits. arXiv:2111.00510. https://doi.org/10.48550/arXiv.2111.00510
11. Lidar D.A., Biham O. Simulating Ising spin glasses on a quantum computer. Physical Review E. 1997;56(3):3661. https://doi.org/10.1103/PhysRevE.56.3661
12. Arute F. et al. Quantum supremacy using a programmable superconducting processor. Nature. 2019;574(7779):505-510. https://doi.org/10.1038/s41586-019-1666-5
13. Zhong H.S. et al. Quantum computational advantage using photons. Science. 2020;370(6523):1460-1463. https://doi.org/10.1126/science.abe8770
14. Wu Y. et al. Strong quantum computational advantage using a superconducting quantum processor. Physical review letters. 2021;127(18);180501. https://doi.org/10.1103/PhysRevLett.127.180501
15. Peruzzo A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications. 2014;5(1):4213. https://doi.org/10.1038/ncomms5213
16. McClean J.R. et al. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics. 2016;18(2):023023. https://doi.org/10.1088/1367-2630/18/2/023023
17. Cao Y., Romero J., Aspuru-Guzik A. Potential of quantum computing for drug discovery. IBM Journal of Research and Development. 2018;62(6):1-20. https://doi.org/10.1147/JRD.2018.2888987
18. Blunt N.S. et al. Perspective on the Current State-of-the-Art of Quantum Computing for Drug Discovery Applications. Journal of Chemical Theory and Computation. 2022;18(12):7001-7023. https://doi.org/10.1021/acs.jctc.2c00574
19. Lordi V., Nichol J.M. Advances and opportunities in materials science for scalable quantum computing. MRS Bulletin. 2021;46:589-595. https://doi.org/10.1557/s43577-021-00133-0
20. Cao Y. et al. Quantum chemistry in the age of quantum computing. Chemical reviews. 2019;119(19):10856-10915. https://doi.org/10.1021/acs.chemrev.8b00803
21. Powell M.J.D. A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation. In: Gomez S., Hennart J.P. (eds.) Advances in Optimization and Numerical Analysis. Mathematics and Its Applications. Vol. 275. Dordrecht: Springer; 1994. p. 51-67. https://doi.org/10.1007/978-94-015-8330-5_4
22. Powell M.J.D. Direct search algorithms for optimization calculations. Acta Numerica. 1998;(7):287-336. https://doi.org/10.1017/S0962492900002841
23. Andreev A.S., Khrapov P.V. Emulators of Quantum Computers on Qubits and on Qudits. ModernInformationTechnologiesandIT-Education. 2022;18(2):455-467. https://doi.org/10.25559/SITITO.18.202202.455-467
24. Khrapov P.V. Cluster expansion and spectrum of the transfer matrix of the two-dimensional Ising model with strong external field. Theoretical and Mathematical Physics. 1984;60(1):734-735. https://doi.org/10.1007/BF01018259
25. Khrapov P.V. Fourier Transform of Transfer Matrices of Plane Ising Models. ModernInformationTechnologiesandIT-Education. 2019;15(2):306-311. https://doi.org/10.25559/SITITO.15.201902.306-311
26. Khrapov P.V. Disorder Solutions for Generalized Ising Model with Multispin Interaction. ModernInformationTechnologiesandIT-Education. 2019;15(2):312-319. https://doi.org/10.25559/SITITO.15.201902.312-319
Published
2023-10-15
How to Cite
ANDREEV, Andrey Sergeevich; KHRAPOV, Pavel Vasilevich. Simulating Ising-Like Models on Quantum Computer. Modern Information Technologies and IT-Education, [S.l.], v. 19, n. 3, p. 554-563, oct. 2023. ISSN 2411-1473. Available at: <http://sitito.cs.msu.ru/index.php/SITITO/article/view/1011>. Date accessed: 02 aug. 2025. doi: https://doi.org/10.25559/SITITO.019.202303.554-563.
Citation Formats
Issue
Vol 19 No 3 (2023): Modern Information Technologies and IT-education
Section
Theoretical Questions of Computer Science, Computer Mathematics
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication policy of the journal is based on traditional ethical principles of the Russian scientific periodicals and is built in terms of ethical norms of editors and publishers work stated in Code of Conduct and Best Practice Guidelines for Journal Editors and Code of Conduct for Journal Publishers, developed by the Committee on Publication Ethics (COPE). In the course of publishing editorial board of the journal is led by international rules for copyright protection, statutory regulations of the Russian Federation as well as international standards of publishing. 
Authors publishing articles in this journal agree to the following: They retain copyright and grant the journal right of first publication of the work, which is automatically licensed under the Creative Commons Attribution License (CC BY license). Users can use, reuse and build upon the material published in this journal provided that such uses are fully attributed.