Quantum algorithm for solving a quadratic nonlinear system of equations
Cheng Xue, Xiao‐Fan Xu, Yu-Chun Wu, Guo‐Ping Guo · Physical Review A · 2022
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 $\ensuremath{\epsilon}$-close to the normalized exact solution of the QNSE with success probability $\mathrm{\ensuremath{\Omega}}(1)$. The complexity of our algorithm is $O(\mathrm{polylog}(n/\ensuremath{\epsilon}))$, which provides an exponential improvement over the optimal classical algorithm in dimension $n$, and the dependence on $\ensuremath{\epsilon}$ 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.