The eigenvalue problem, a cornerstone in linear algebra, provides profound insights into studying matrix properties. Quantum algorithms addressing this problem have hitherto been constrained to special normal matrices assuming spectral decomposition, leaving the extension to general matrices an open challenge. In this work, we present a novel family of quantum algorithms tailored for solving the eigenvalue problem for general matrices, encompassing scenarios with complex eigenvalues or even defective matrices. Our approach begins by tackling the task of searching for an eigenvalue without additional constraints. For diagonalizable matrices, our algorithm has $\tilde O(\varepsilon^{-1})$ complexity with an error $\varepsilon$, achieving the nearly Heisenberg scaling. Subsequently, we study the identification of eigenvalues closest to a specified point or line, extending the results for ground energy and energy gap problems in Hermitian matrices. We achieve an accuracy scaling of $\tilde O(\varepsilon^{-2})$ for general diagonalizable matrices, further refining to $\tilde O(\varepsilon^{-1})$ under the condition of real eigenvalues or constant distance from the reference point. The algorithm's foundation lies in the synergy of three techniques: the relationship between eigenvalues of matrix $A$ and the minimum singular value of $A-\mu I$, quantum singular value threshold subroutine extended from quantum singular-value estimation, and problem-specific searching algorithms. Our algorithms find applications in diverse domains, including estimating the relaxation time of Markov chains, solving Liouvillian gaps in open quantum systems, and verifying PT-symmetry broken/unbroken phases. These applications underscore the significance of our quantum eigensolvers for problems across various disciplines.

翻译：特征值问题是线性代数中的基石，为研究矩阵性质提供了深刻见解。以往针对该问题的量子算法受限于假设谱分解的特殊正规矩阵，将其扩展至一般矩阵仍是一个开放挑战。在本工作中，我们提出了一系列新颖的量子算法，专门用于求解一般矩阵（包括具有复特征值甚至亏损矩阵的情形）的特征值问题。我们的方法首先从无附加约束条件下搜索单个特征值入手。对于可对角化矩阵，算法具有$\tilde O(\varepsilon^{-1})$的复杂度，误差为$\varepsilon$，实现了近海森堡标度。随后，我们研究了识别最接近指定点或直线的特征值问题，将厄米矩阵中基态能量和能隙问题的结果进行了扩展。对于一般可对角化矩阵，实现了$\tilde O(\varepsilon^{-2})$的精度标度，在特征值为实数或与参考点距离恒定的条件下进一步优化至$\tilde O(\varepsilon^{-1})$。该算法的基础源于三种技术的协同：矩阵$A$的特征值与$A-\mu I$最小奇异值之间的关系、基于量子奇异值估计扩展而来的量子奇异值阈值子程序，以及问题特定的搜索算法。我们的算法适用于多个领域，包括估计马尔可夫链的弛豫时间、求解开放量子系统的李乌维尔间隙，以及验证PT对称性的破缺/未破缺相。这些应用凸显了我们的量子本征值求解器对于跨学科问题的重要意义。