The Krylov subspace methods, being one category of the most important classical numerical methods for linear algebra problems, can be much more powerful when generalised to quantum computing. However, quantum Krylov subspace algorithms are prone to errors due to inevitable statistical fluctuations in quantum measurements. To address this problem, we develop a general theoretical framework to analyse the statistical error and measurement cost. Based on the framework, we propose a quantum algorithm to construct the Hamiltonian-power Krylov subspace that can minimise the measurement cost. In our algorithm, the product of power and Gaussian functions of the Hamiltonian is expressed as an integral of the real-time evolution, such that it can be evaluated on a quantum computer. We compare our algorithm with other established quantum Krylov subspace algorithms in solving two prominent examples. To achieve an error comparable to that of the classical Lanczos algorithm at the same subspace dimension, our algorithm typically requires orders of magnitude fewer measurements than others. Such an improvement can be attributed to the reduced cost of composing projectors onto the ground state. These results show that our algorithm is exceptionally robust to statistical fluctuations and promising for practical applications.

中文翻译：

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测量高效的量子克雷洛夫子空间对角化


Krylov 子空间方法是线性代数问题最重要的经典数值方法之一，当推广到量子计算时，它的功能会更加强大。然而，由于量子测量中不可避免的统计波动，量子克雷洛夫子空间算法很容易出错。为了解决这个问题，我们开发了一个通用的理论框架来分析统计误差和测量成本。基于该框架，我们提出了一种量子算法来构造哈密顿幂克雷洛夫子空间，可以最小化测量成本。在我们的算法中，哈密顿量的幂和高斯函数的乘积被表示为实时演化的积分，以便可以在量子计算机上对其进行评估。我们将我们的算法与其他已建立的量子 Krylov 子空间算法在解决两个突出例子方面进行比较。为了在相同的子空间维度上实现与经典 Lanczos 算法相当的误差，我们的算法通常需要比其他算法少几个数量级的测量。这种改进可以归因于将投影仪组合到基态上的成本降低。这些结果表明，我们的算法对统计波动非常稳健，并且具有实际应用前景。