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。以往的量子算法通过将偏微分方程离散化求解线性代数方程组，其计算复杂度至少与计算中所涉矩阵的条件数 $\kappa$ 呈线性标度。对于许多实际应用而言，$\kappa$ 随矩阵规模 $N$ 呈多项式标度，这导致这些算法的复杂度为 $N$ 的多项式函数。本文提出一种量子算法，其复杂度为 $N$ 的多对数函数，且对一大类偏微分方程而言与 $\kappa$ 无关。我们的算法生成的量子态可提取解的特征。方法论的核心在于采用小波基作为辅助坐标系，通过简单的对角预处理器使得相关矩阵的条件数与 $N$ 无关。我们通过数值模拟展示了小波预处理对多个微分方程的影响。本工作可为采用标准离散化方法的量子模拟算法性能提升提供实用途径。