Quantum Algorithm for Solving a Quadratic Nonlinear System of Equations

• URL: http://arxiv.org/abs/2112.01655v3
• Date: Sat, 8 Oct 2022 07:44:12 GMT
• Title: Quantum Algorithm for Solving a Quadratic Nonlinear System of Equations
• Authors: Cheng Xue, Xiao-Fan Xu, Yu-Chun Wu, Guo-Ping Guo
• Abstract summary: The complexity of our algorithm is $O(rm polylog(n/epsilon))$, which provides an exponential improvement over the optimal classical algorithm in dimension $n$. Our algorithm exponentially accelerates the solution of QNSE and has wide applications in all kinds of nonlinear problems.
• Score: 0.22940141855172036
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Solving a quadratic nonlinear system of equations (QNSE) is a fundamental, but important, task in nonlinear science. We propose an efficient quantum algorithm for solving $n$-dimensional QNSE. Our algorithm embeds QNSE into a finite-dimensional system of linear equations using the homotopy perturbation method and a linearization technique; then we solve the linear equations with a quantum linear system solver and obtain a state which is $\epsilon$-close to the normalized exact solution of the QNSE with success probability $\Omega(1)$. The complexity of our algorithm is $O({\rm polylog}(n/\epsilon))$, which provides an exponential improvement over the optimal classical algorithm in dimension $n$, and the dependence on $\epsilon$ is almost optimal. Therefore, our algorithm exponentially accelerates the solution of QNSE and has wide applications in all kinds of nonlinear problems, contributing to the research progress of nonlinear science.

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