Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910652490055680 |
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| author | Guzman-Cajica, Francisco Guzman, Francisco S. |
| author_facet | Guzman-Cajica, Francisco Guzman, Francisco S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems Guzman-Cajica, Francisco Guzman, Francisco S. Quantum Physics Computational Physics 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. |
| title | Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems |
| topic | Quantum Physics Computational Physics |
| url | https://arxiv.org/abs/2404.07016 |