Fast quantum algorithm for differential equations

TL;DR

This paper introduces a quantum algorithm for solving certain PDEs with complexity polylogarithmic in matrix size and independent of the condition number, using wavelet preconditioning to improve efficiency.

Contribution

The authors develop a quantum algorithm that achieves polylogarithmic complexity in matrix size and is independent of the condition number for a class of PDEs, utilizing wavelet basis preconditioning.

Findings

Numerical simulations demonstrate the effectiveness of wavelet preconditioning.

The algorithm achieves complexity polylogarithmic in matrix size, independent of condition number.

Potential for practical quantum simulation improvements.

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.