Abstract
The power method (or iteration) is a well-known classical technique that can be used to find the dominant eigenpair of a matrix. Here, we present a variational quantum circuit method for the power iteration, which can be used to find the eigenpairs of unitary matrices and so their associated Hamiltonians. We discuss how to apply the circuit to combinatorial optimization problems formulated as a quadratic unconstrained binary optimization and discuss its complexity. In addition, we run numerical simulations for random problem instances with up to 21 parameters and observe that the method can generate solutions to the optimization problems with only a few number of iterations and the growth in the number of iterations is polynomial in the number of parameters. Therefore, the circuit can be simulated on the near-term quantum computers with ease.
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Notes
You can access the simulation code for the variational quantum power method for random quadratic binary optimization from the link: https://github.com/adaskin/vqpm_public
References
Albash, T., Lidar, D.A.: Adiabatic quantum computation. Rev. Modern Phys. 90(1), 015002 (2018)
Braunstein, S.L., Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72(22), 3439 (1994)
Christandl, M., Renner, R.: Reliable quantum state tomography. Phys. Rev. Lett. 109(12), 120403 (2012)
Cramér, H.: Mathematical Methods of Statistics, vol. 43. Princeton University Press, Princeton (1999)
Daskin, A. (2020) The quantum version of the shifted power method and its application in quadratic binary optimization. Turk J Elec Eng and Comp Sci, http://arxiv.org/abs/1809.01378
Ezugwu, A.E., Pillay, V., Hirasen, D., Sivanarain, K., Govender, M.: A comparative study of meta-heuristic optimization algorithms for 0–1 knapsack problem: Some initial results. IEEE Access 7, 43979–44001 (2019)
Farhi, E., Goldstone, J., Gutmann, S., Sipser, M. (2000). Quantum computation by adiabatic evolution. http://arxiv.org/abs/quant-ph/0001106
Glover, F., Kochenberger, G., Du, Y.: Quantum bridge analytics i: a tutorial on formulating and using qubo models. 4OR 17(4), 335–371 (2019)
Hentschel, A., Sanders, B.C.: Machine learning for precise quantum measurement. Phys. Rev. Lett. 104(6), 063603 (2010)
Hou, Z., Zhu, H., Xiang, G.Y., Li, C.F., Guo, G.C.: Achieving quantum precision limit in adaptive qubit state tomography. npj Quantum Inf. 2(1), 1–5 (2016)
Korte, B., Vygen, J.: Combinatorial Optimization, vol. 2. Springer, Berlin (2012)
Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308–313 (1965)
Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.H., Zhou, X.Q., Love, P.J., Aspuru-Guzik, A., Obrien, J.L.: A variational eigenvalue solver on a photonic quantum processor. Nature Commun. 5, 4213 (2014)
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Daskin, A. Combinatorial optimization through variational quantum power method. Quantum Inf Process 20, 336 (2021). https://doi.org/10.1007/s11128-021-03283-x
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DOI: https://doi.org/10.1007/s11128-021-03283-x