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-in-$N$ complexity 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 that enables extracting features of the solution. Central to our methodology is using a wavelet basis as an auxiliary system of coordinates in which the condition number of associated matrices is 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.

翻译：偏微分方程在科学与工程中无处不在。现有的量子算法通过离散化偏微分方程得到线性代数方程组，其计算复杂度通常与计算过程中矩阵的条件数 $\kappa$ 至少呈线性关系。在许多实际应用中，$\kappa$ 随矩阵规模 $N$ 呈多项式增长，导致算法复杂度为 $N$ 的多项式量级。本文针对一大类偏微分方程提出一种量子算法，其复杂度在 $N$ 上呈多对数级别，且与 $\kappa$ 无关。该算法生成的量子态能够提取解的特征信息。方法论的核心是将小波基作为辅助坐标系，通过简单的对角线预处理器使相关矩阵的条件数不依赖于 $N$ 。我们针对多个微分方程进行了数值模拟，展示了小波预处理器的效果。这项工作可为采用标准离散化方法的量子模拟算法提供实际性能提升途径。