Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity $\mathrm{poly}(1/\epsilon)$, where $\epsilon$ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be $\mathrm{poly}(d, \log(1/\epsilon))$, where $d$ is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.

中文翻译：

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偏微分方程的高精度量子算法

量子计算机可以对微分方程组的解产生量子编码，比经典算法可以产生显式描述的速度快得多。然而，虽然线性常微分方程的高精度量子算法已经很好地建立，但线性偏微分方程 (PDE) 的最佳量子算法具有复杂性 $\mathrm{poly}(1/\epsilon)$，其中 $\epsilon $ 是容错性。通过开发基于自适应阶次有限差分方法和谱方法的量子算法，我们将线性偏微分方程的量子算法复杂度提高到 $\mathrm{poly}(d,\log(1/\epsilon))$，其中 $ d$ 是空间维度。我们的算法将高精度量子线性系统算法应用于我们限制条件数和近似误差的系统。我们开发了泊松方程的有限差分算法和更一般的二阶椭圆方程的谱算法。