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arXiv · 2404.07016

Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems

Francisco Guzman-Cajica·Francisco S. Guzman
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Abstract

In this paper we use a Variational Quantum Algorithm to solve Initial Value Problems with the Implicit Crank-Nicolson and the Method of Lines (MoL) evolution schemes. The unknown functions use a spectral decomposition with the Fourier basis. The examples developed to illustrate the implementation are the Advection equation, the wave equation written as a system of first order coupled equations and the viscous Burgers equation as a non-linear case. The problems are solved using: i) standard Finite Differences as the solution to compare with, ii) the State Vector Formalism (SVF), and iii) the Sampling Error Formalism (SEF). The contributions of this paper include: 1) cost functions for generic first order in time PDEs using the implicit Crank-Nicholson and the MoL, 2) detailed convergence or self-convergence tests are presented for all the equations solved, 3) a system of three coupled PDEs is solved, 4) solutions using sampling error are presented, which highlights the importance of simulating the sampling process and 5) a fast version of the SVF and SEF was developed which can be used to test different optimizers faster.

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Francisco Guzman-Cajica, Francisco S. Guzman. 2024-04-10. Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems. https://doi.org/10.1103/physreva.110.042415

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