求稳定厄米矩阵幂的量子算法
Quantum algorithms for powering stable Hermitian matrices
摘要：
矩阵幂是线性代数中一个基本的计算本原。它在科学计算和工程中有着广泛的应用，并且是时间齐次线性常微分方程的解，离散时间马尔可夫链的模拟或用迭代方法发现矩阵的谱性质的基础。本文研究了在量子计算机上加速稀疏稳定厄米矩阵的矩阵赋能的可能性。本文提出了两种量子算法，它们可以在经典的矩阵赋能算法上获得加速比：（i）一种基于量子行走的快速转发算法（ii）一种基于Hamilton模拟的算法。此外，通过将N比特奇偶校验确定问题映射到矩阵供电问题，我们提供了限制在供电非厄米矩阵中可实现的量子加速比的不可行定理。
英文摘要：
Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering, and underlies the solution of time-homogeneous linear ordinary differential equations, simulation of discrete-time Markov chains, or discovering the spectral properties of matrices with iterative methods. In this paper, we investigate the possibility of speeding up matrix powering of sparse stable Hermitian matrices on a quantum computer. We present two quantum algorithms that can achieve speedup over the classical matrix powering algorithms -- (i) an adaption of quantum-walk based fast forwarding algorithm (ii) an algorithm based on Hamiltonian simulation. Furthermore, by mapping the N-bit parity determination problem to a matrix powering problem, we provide no-go theorems that limit the quantum speedups achievable in powering non-Hermitian matrices.
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