Quantum subspace diagonalization methods are an exciting new class of algorithms for solving large\rev{-}scale eigenvalue problems using quantum computers. Unfortunately, these methods require the solution of an ill-conditioned generalized eigenvalue problem, with a matrix pair corrupted by a non-negligible amount of noise that is far above the machine precision. Despite pessimistic predictions from classical \rev{worst-case} perturbation theories, these methods can perform reliably well if the generalized eigenvalue problem is solved using a standard truncation strategy. By leveraging and advancing classical results in matrix perturbation theory, we provide a theoretical analysis of this surprising phenomenon, proving that under certain natural conditions, a quantum subspace diagonalization algorithm can accurately compute the smallest eigenvalue of a large Hermitian matrix. We give numerical experiments demonstrating the effectiveness of the theory and providing practical guidance for the choice of truncation level. Our new results can also be of independent interest to solving eigenvalue problems outside the context of quantum computation.

翻译：量子子空间对角化方法是一类利用量子计算机解决大规模本征值问题的新兴算法。然而，这类方法需解决一个病态广义本征值问题，其矩阵对受到的噪声干扰远超机器精度，且不可忽略。尽管经典最坏情况扰动理论给出悲观预测，但若采用标准截断策略求解该广义本征值问题，这些方法仍能可靠运行。通过借鉴并发展矩阵扰动理论中的经典结论，我们对此反常现象进行了理论分析，证明在特定自然条件下，量子子空间对角化算法可精确计算大型埃尔米特矩阵的最小本征值。我们通过数值实验验证了该理论的有效性，并为截断水平的选择提供了实践指导。我们的新结论在量子计算领域之外的本征值问题求解中亦具有独立参考价值。