OPTIMIZING QUANTUM ALGORITHMS FOR SOLVING THE POISSON EQUATION
DOI:
https://doi.org/10.37943/18REAT9767Keywords:
partial differential equation, poisson equation, quantum computing, variational quantum eigensolver, optimizationAbstract
Contemporary quantum computers open up novel possibilities for tackling intricate problems, encompassing quantum system modeling and solving partial differential equations (PDEs). This paper explores the optimization of quantum algorithms aimed at resolving PDEs, presenting a significant challenge within the realm of computational science. The work delves into the application of the Variational Quantum Eigensolver (VQE) for addressing equations such as Poisson's equation. It employs a Hamiltonian constructed using a modified Feynman-Kitaev formalism for a VQE, which represents a quantum system and encapsulates information pertaining to the classical system. By optimizing the parameters of the quantum circuit that implements this Hamiltonian, it becomes feasible to achieve minimization, which corresponds to the solution of the original classical system. The modification optimizes quantum circuits by minimizing the cost function associated with the VQE. The efficacy of this approach is demonstrated through the illustrative example of solving the Poisson equation. The prospects for its application to the integration of more generalized PDEs are discussed in detail. This study provides an in-depth analysis of the potential advantages of quantum algorithms in the domain of numerical solutions for the Poisson equation and emphasizes the significance of continued research in this direction. By leveraging quantum computing capabilities, the development of more efficient methodologies for solving these equations is possible, which could significantly transform current computational practices. The findings of this work underscore not only the practical advantages but also the transformative potential of quantum computing in addressing complex PDEs. Moreover, the results obtained highlight the critical need for ongoing research to refine these techniques and extend their applicability to a broader class of PDEs, ultimately paving the way for advancements in various scientific and engineering domains.
References
Tosti Balducci, G., Chen, B., Möller, M., Gerritsma, M., & De Breuker, R. (2022). Review and perspectives in quantum computing for partial differential equations in structural mechanics. In Frontiers in Mechanical Engineering, 8. Frontiers Media SA. https://doi.org/10.3389/fmech.2022.914241
Saha, K. K., Robson, W., Howington, C., Suh, I.-S., Wang, Z., & Nabrzyski, J. (2022). Advancing Algorithm to Scale and Accurately Solve Quantum Poisson Equation on Near-term Quantum Hardware (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2210.16668
Wang, S., Wang, Z., Li, W., Fan, L., Wei, Z., & Gu, Y. (2020). Quantum fast Poisson solver: the algorithm and complete and modular circuit design. In Quantum Information Processing, 19(6). Springer Science and Business Media LLC. https://doi.org/10.1007/s11128-020-02669-7
Leong, F. Y., Koh, D. E., Ewe, W.-B., & Kong, J. F. (2023). Variational quantum simulation of partial differential equations: applications in colloidal transport. In International Journal of Numerical Methods for Heat & Fluid Flow, 33(11), 3669–3690. Emerald. https://doi.org/10.1108/hff-05-2023-0265
Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S. C., Endo, S., Fujii, K., McClean, J. R., Mitarai, K., Yuan, X., Cincio, L., & Coles, P. J. (2021). Variational quantum algorithms. In Nature Reviews Physics, 3(9), 625–644. Springer Science and Business Media LLC. https://doi.org/10.1038/s42254-021-00348-9
Cao, Y., Papageorgiou, A., Petras, I., Traub, J., & Kais, S. (2013). Quantum algorithm and circuit design solving the Poisson equation. In New Journal of Physics 15(1), 013021. IOP Publishing. https://doi.org/10.1088/1367-2630/15/1/013021
Liu, H.-L., Wu, Y.-S., Wan, L.-C., Pan, S.-J., Qin, S.-J., Gao, F., & Wen, Q.-Y. (2021). Variational quantum algorithm for the Poisson equation. In Physical Review A, 104(2). American Physical Society (APS). https://doi.org/10.1103/physreva.104.022418
Sato, Y., Kondo, R., Koide, S., Takamatsu, H., & Imoto, N. (2021). Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation. In Physical Review A, 104(5). American Physical Society (APS). https://doi.org/10.1103/physreva.104.052409
Daribayev, B., Mukhanbet, A., & Imankulov, T. (2023). Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators. In Applied Sciences, 13, (20), 11491. MDPI AG. https://doi.org/10.3390/app132011491
Li, H.-M., Wang, Z.-X., & Fei, S.-M. (2023). Variational quantum algorithms for Poisson equations based on the decomposition of sparse Hamiltonians. In Physical Review A, 108(3). American Physical Society (APS). https://doi.org/10.1103/physreva.108.032418
Ali, M., & Kabel, M. (2023). Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer. In Physical Review Applied, 20 (1). American Physical Society (APS). https://doi.org/10.1103/physrevapplied.20.014054
Bonet-Monroig, X., Wang, H., Vermetten, D., Senjean, B., Moussa, C., Bäck, T., Dunjko, V., & O’Brien, T. E. (2023). Performance comparison of optimization methods on variational quantum algorithms. In Physical Review A, 107(3). American Physical Society (APS). https://doi.org/10.1103/physreva.107.032407
Cui, G., Wang, Z., Wang, S., Shi, S., Shang, R., Li, W., Wei, Z., & Gu, Y. (2021). Optimization and Noise Analysis of the Quantum Algorithm for Solving One-Dimensional Poisson Equation. arXiv. https://doi.org/10.48550/ARXIV.2108.12203
Soloviev, V. P., Larrañaga, P., & Bielza, C. (2022). Quantum parametric circuit optimization with estimation of distribution algorithms. In Proceedings of the Genetic and Evolutionary Computation Conference Companion. GECCO ’22: Genetic and Evolutionary Computation Conference. ACM. https://doi.org/10.1145/3520304.3533963
Ravi, G. S., Smith, K. N., Gokhale, P., Mari, A., Earnest, N., Javadi-Abhari, A., & Chong, F. T. (2021). VAQEM: A Variational Approach to Quantum Error Mitigation (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2112.05821
Khan, A., Clark, B. K., & Tubman, N. M. (2023). Pre-optimizing variational quantum eigensolvers with tensor networks (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2310.12965
de Keijzer, R. J. P. T., Colussi, V. E., Škorić, B., & Kokkelmans, S. J. J. M. F. (2022). Optimization of the variational quantum eigensolver for quantum chemistry applications. In AVS Quantum Science, 4(1). American Vacuum Society. https://doi.org/10.1116/5.0076435
Nicoli, K., Anders, C. J., Funcke, L., Hartung, T., Jansen, K., Kühn, S., Müller, K.-R., Stornati, P., Kessel, P., & Nakajima, S. (2023). Physics-Informed Bayesian Optimization of Variational Quantum Circuits. In Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track.
Endo, K., Sato, Y., Raymond, R., Wada, K., Yamamoto, N., & Watanabe, H. C. (2023). Optimal Parameter Configurations for Sequential Optimization of Variational Quantum Eigensolver. arXiv. https://doi.org/10.48550/ARXIV.2303.07082
Hahm, J., Kim, H., & Park, Y. J. (2023). Improvement in Variational Quantum Algorithms by Measurement Simplification (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2312.06176
Tilly, J., Chen, H., Cao, S., Picozzi, D., Setia, K., Li, Y., Grant, E., Wossnig, L., Rungger, I., Booth, G. H., & Tennyson, J. (2022). The Variational Quantum Eigensolver: A review of methods and best practices. In Physics Reports, 986, 1–128. Elsevier BV. https://doi.org/10.1016/j.physrep.2022.08.003
Gottesman, D., Kitaev, A. Yu., Shen, A. H., & Vyalyi, M. N. (2003). Classical and Quantum Computation. In The American Mathematical Monthly, 110(10), 969. JSTOR. https://doi.org/10.2307/3647986
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