Quantum Krylov subspace diagonalization (QKSD) is an emerging method used in place of quantum phase estimation in the early fault-tolerant era, where limited quantum circuit depth is available. In contrast to the classical Krylov subspace diagonalization (KSD) or the Lanczos method, QKSD exploits the quantum computer to efficiently estimate the eigenvalues of large-size Hamiltonians through a faster Krylov projection. However, unlike classical KSD, which is solely concerned with machine precision, QKSD is inherently accompanied by errors originating from a finite number of samples. Moreover, due to difficulty establishing an artificial orthogonal basis, ill-conditioning problems are often encountered, rendering the solution vulnerable to noise. In this work, we present a nonasymptotic theoretical framework to assess the relationship between sampling noise and its effects on eigenvalues. We also propose an optimal solution to cope with large condition numbers by eliminating the ill-conditioned bases. Numerical simulations of the one-dimensional Hubbard model demonstrate that the error bound of finite samplings accurately predicts the experimental errors in well-conditioned regions.

中文翻译：

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量子Krylov子空间对角化中的采样误差分析


量子克雷洛夫子空间对角化（QKSD）是一种新兴方法，用于在早期容错时代代替量子相位估计，当时可用的量子电路深度有限。与经典的 Krylov 子空间对角化 (KSD) 或 Lanczos 方法相比，QKSD 利用量子计算机通过更快的 Krylov 投影来有效估计大尺寸哈密顿量的特征值。然而，与仅关注机器精度的经典 KSD 不同，QKSD 本质上伴随着源自有限数量样本的误差。此外，由于建立人工正交基的困难，经常遇到病态问题，使得解决方案容易受到噪声的影响。在这项工作中，我们提出了一个非渐近理论框架来评估采样噪声及其对特征值的影响之间的关系。我们还提出了一种通过消除病态碱基来应对大条件数的最佳解决方案。一维哈伯德模型的数值模拟表明，有限采样的误差界准确地预测了条件良好区域的实验误差。