While non-Hermitian physics has attracted considerable attention, current studies are limited to small or classically solvable systems. Quantum computing, as a powerful eigensolver, have predominantly been applied to Hermitian domain, leaving their potential for studying non-Hermitian problems largely unexplored. We extend the power of quantum computing to general non-Hermitian eigenproblems. Our approach works for finding eigenvalues without extra constrains, or eigenvalues closest to specified points or lines, thus extending results for ground energy and energy gap problems for Hermitian matrices. Our algorithms have broad applications, and as examples, we consider two central problems in non-Hermitian physics. Firstly, our approach is the first to offer an efficient quantum solution to the witness of spontaneous $PT$-symmetry breaking, and provide provable, exponential quantum advantage. Secondly, our approach enables the estimation of Liouvillian gap, which is crucial for characterizing relaxation times. Our general approach can also find applications in many other areas, such as the study of Markovian stochastic processes. These results underscore the significance of our quantum algorithms for addressing practical eigenproblems across various disciplines.

翻译：尽管非厄米物理已引起广泛关注，但现有研究仅限于小型或经典可解系统。量子计算作为一种强大的特征求解器，目前主要应用于厄米领域，其在非厄米问题研究中的潜力尚未得到充分探索。我们将量子计算的能力拓展至一般非厄米特征值问题。我们的方法适用于求解无额外约束的特征值，或最接近指定点或线的特征值，从而将厄米矩阵基态能量与能隙问题的研究成果进行推广。所提算法具有广泛适用性，我们以非厄米物理中的两个核心问题为例进行说明。首先，我们的方法首次为自发$PT$对称破缺的见证提供了高效的量子解决方案，并给出可证明的指数级量子优势。其次，该方法能够估计对弛豫时间表征至关重要的刘维尔间隙。我们的通用方法还可应用于许多其他领域，例如马尔可夫随机过程的研究。这些结果凸显了我们的量子算法在解决跨学科实际特征值问题方面的重要意义。