Time Complexity Analysis of Quantum Algorithms Via Linear Representations For Nonlinear Ordinary and Partial Differential Equations

Download This Paper Open PDF in Browser Add Paper to My Library Share: Permalink Using these links will ensure access to this page indefinitely Copy URL Time Complexity Analysis of Quantum Algorithms Via Linear Representations For Nonlinear Ordinary and Partial Differential Equations 60 Pages Posted: 3 Oct 2022 See all articles by Shi JinShi Jinaffiliation not provided to SSRNNana Liuaffiliation not provided to SSRNYue Yuaffiliation not provided to SSRN Abstract We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrodinger framework to solve the Liouville equation for the HJE, since it can be recast as the semiclassical limit of the Wigner transform of the Schrodinger equation. Comparsion between the Schrodinger and the Liouville framework will also be made. Keywords: Linear representation methods, Liouville representation, Koopman-von Neumann representation, Semiclassical Schrodinger equation, Quantum linear systems algorithms, Quantum simulation Suggested Citation: Suggested Citation Jin, Shi and Liu, Nana and Yu, Yue, Time Complexity Analysis of Quantum Algorithms Via Linear Representations For Nonlinear Ordinary and Partial Differential Equations. Available at SSRN: https://ssrn.com/abstract=4236129 Shi Jin affiliation not provided to SSRN ( email ) No Address Available Nana Liu affiliation not provided to SSRN ( email ) No Address Available Yue Yu (Contact Author) affiliation not provided to SSRN ( email ) No Address Available Download This Paper Open PDF in Browser Do you have a job opening that you would like to promote on SSRN? Place Job Opening Paper statistics Downloads 2 Abstract Views 6 PlumX Metrics Feedback Feedback to SSRN Feedback (required) Email (required) Submit If you need immediate assistance, call 877-SSRNHelp (877 777 6435) in the United States, or +1 212 448 2500 outside of the United States, 8:30AM to 6:00PM U.S. Eastern, Monday - Friday. Submit a Paper Section 508 Text Only Pages SSRN Quick Links SSRN Solutions Research Paper Series Conference Papers Partners in Publishing Jobs & Announcements Newsletter Sign Up SSRN Rankings Top Papers Top Authors Top Organizations About SSRN SSRN Objectives Network Directors Presidential Letter Announcements Contact us FAQs Copyright Terms and Conditions Privacy Policy We use cookies to help provide and enhance our service and tailor content. To learn more, visit Cookie Settings. This page was processed by aws-apollo-5dc in 0.266 seconds