量子斜特征投影
Quantum oblique eigenprojection
- AWS Center for Quantum Computing（亚马逊网络服务量子计算中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究提出一种量子斜特征投影算法，利用块编码和离散傅里叶变换实现非正交特征子空间分解，在复特征值、Riccati方程及Sylvester方程求解上实现显著加速。
AI中文摘要：
每个方阵将其底层希尔伯特空间分解为广义的非正交特征子空间。我们证明，在给定输入矩阵的块编码访问权限的情况下，量子计算机能够执行这种斜特征投影 $\Pi$。我们的方法在输入满足谱集条件时，其查询复杂度在逆间隙上接近线性，且归一化因子接近 $\lVert\Pi\rVert$。这涵盖了关于数值范围或可对角化性的常见假设，并与正交特征投影的已知结果相匹配。我们通过使用矩阵预解式的离散傅里叶变换进行双侧块预处理来实现这一点。我们描述了以下应用：（i）制备具有复特征值的矩阵的特征态，将Low和Su的量子特征值变换算法扩展到实谱之外；（ii）求解连续时间代数Riccati方程，将Rodenas-Ruiz、Zhao和Lee先前求解器的速度提升三次方；（iii）求解普通Sylvester方程，对Wang和Liu的直接增广方法进行二次改进。我们的结果表明，在量子计算机上应用非解析矩阵函数是一条有前景的途径。
英文摘要：
Every square matrix decomposes its underlying Hilbert space into generalized, nonorthogonal eigensubspaces. We show that a quantum computer can perform such an oblique eigenprojection $Π$ given block encoding access to the input matrix. Our approach has a query complexity nearly linear in the inverse gap and a normalization factor close to $\lVertΠ\rVert$ under a spectral-set condition on the input. This covers common assumptions on the numerical range or diagonalizability and matches known results for orthogonal eigenprojections. We achieve this with a two-sided block preconditioning that uses a discrete Fourier transform of the matrix resolvent. We describe applications to: (i) preparing eigenstates of matrices with complex eigenvalues, extending the quantum eigenvalue transformation algorithm of Low and Su beyond real spectra; (ii) solving continuous-time algebraic Riccati equations, cubically speeding up a prior solver of Rodenas-Ruiz, Zhao, and Lee; and (iii) solving ordinary Sylvester equations, quadratically improving a direct augmented method of Wang and Liu. Our result suggests a promising route to applying nonanalytic matrix functions on quantum computers.
