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arXiv · 2306.11802

Fast quantum algorithm for differential equations

Mohsen Bagherimehrab·Kouhei Nakaji·Nathan Wiebe·Gavin K. Brennen·Barry C. Sanders·Alán Aspuru-Guzik
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Abstract

Partial differential equations (PDEs) are ubiquitous in science and engineering. Prior quantum algorithms for solving the system of linear algebraic equations obtained from discretizing a PDE have a computational complexity that scales at least linearly with the condition number $\kappa$ of the matrices involved in the computation. For many practical applications, $\kappa$ scales polynomially with the size $N$ of the matrices, rendering a polynomial complexity in $N$ for these algorithms. Here we present a quantum algorithm with a complexity that is polylogarithmic in $N$ but is independent of $\kappa$ for a large class of PDEs. Our algorithm generates a quantum state from which features of the solution can be extracted. Central to our methodology is using a wavelet basis as an auxiliary system of coordinates in which the condition number of associated matrices becomes independent of $N$ by a simple diagonal preconditioner. We present numerical simulations showing the effect of the wavelet preconditioner for several differential equations. Our work could provide a practical way to boost the performance of quantum simulation algorithms where standard methods are used for discretization.

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Mohsen Bagherimehrab, Kouhei Nakaji, Nathan Wiebe, Gavin K. Brennen, Barry C. Sanders, Alán Aspuru-Guzik. 2023-06-20. Fast quantum algorithm for differential equations. https://arxiv.org/abs/2306.11802

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