AI 中文总结
研究旨在改进量子多体哈密顿量低能谱计算方法。提出正交量子克里洛夫子空间对角化（OQKD）框架，通过算子层面的兰索斯递归实现正交量子化，无需重叠矩阵正则化，还引入重启态制备协议，提升了算法性能与稳定性。
AI 中文摘要
量子子空间对角化方法，特别是量子克里洛夫子空间对角化（QKD），为计算量子多体哈密顿量的低能谱提供了一条有前景的途径。然而，现有的量子克里洛夫方法依赖非正交克里洛夫子空间，需要重叠矩阵正则化，这限制了数值稳定性和准确性。在这项工作中，我们引入了正交量子克里洛夫子空间对角化（OQKD）框架，在算子层面重新构建经典的兰索斯递归，实现克里洛夫子空间对角化的正交量子实现。通过将兰索斯向量表示为哈密顿量的多项式变换，OQKD重现了经典兰索斯算法的正交性、三对角结构和收敛行为，从而无需重叠矩阵正则化。我们进一步表明，所需的兰索斯多项式可以使用块编码和广义量子信号处理来实现，其渐近查询复杂度与基于切比雪夫的QKD方法相同。对J1 - J2海森堡模型的数值模拟证实了该方法的经典兰索斯收敛性和数值稳定性，同时通过解析建立了测量复杂度缩放。基于OQKD框架，我们引入了一种重启态制备协议，用一系列固定的低阶变换取代单个高阶多项式变换，在保持可承受的块编码成功概率的同时保持可比的收敛性。这些结果确立了OQKD作为经典兰索斯算法的正交量子模拟，并确定重启协议是量子相位估计中有前景的态制备策略。
英文摘要
Quantum subspace-diagonalization methods, particularly Quantum Krylov Diagonalization (QKD), provide a promising route for computing low-energy spectra of quantum many-body Hamiltonians. However, existing quantum Krylov approaches rely on non-orthogonal Krylov bases, requiring overlap-matrix regularization that limits numerical stability and accuracy. In this work, we introduce an Orthogonal Quantum Krylov Diagonalization (OQKD) framework that reformulates the classical Lanczos recursion at the operator level, enabling an orthogonal quantum implementation of Krylov-subspace diagonalization. By expressing Lanczos vectors as polynomial transformations of the Hamiltonian, OQKD reproduces the orthogonality, tridiagonal structure, and convergence behavior of the classical Lanczos algorithm thus eliminating the need for overlap-matrix regularization. We further show that the required Lanczos polynomials can be implemented using block encoding and Generalized Quantum Signal Processing with the same asymptotic query complexity as Chebyshev-based QKD methods. Numerical simulations of the $J_1$--$J_2$ Heisenberg model confirm the classical Lanczos convergence and numerical stability of the proposed method, while the measurement-complexity scaling is established analytically. Building upon the OQKD framework, we then introduce a restarted state-preparation protocol that replaces a single high-degree polynomial transformation with a sequence of fixed low-degree transformations, maintaining an affordable block encoding success probability while retaining comparable convergence. These results establish OQKD as an orthogonal quantum analog of the classical Lanczos algorithm and identify the restarted protocol as a promising state-preparation strategy for Quantum Phase Estimation.
