发表机构
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN; Department of Electrical and Computer Engineering, University of British Columbia; Department of Physics, University of Guelph; Center for Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology; Department of Physics and Astronomy, University of British Columbia; Stewart Blusson Quantum Matter Institute, University of British Columbia; KEK Theory Center, Institute of Particle and Nuclear Studies; Graduate University for Advanced Studies (SOKENDAI)（RIKEN 跨学科理论与科学中心 (iTHEMS); 不列颠哥伦比亚大学电气与计算机工程系; 圭尔夫大学物理系; 日本海洋地球科学技术局数学科学与先进技术中心; 不列颠哥伦比亚大学物理与天文学系; 不列颠哥伦比亚大学斯图尔特·布拉斯顿量子物质研究所; 高能加速器研究机构理论中心粒子与核研究部; 综合研究大学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
量子Krylov学习（QKL）是一种受限访问框架，通过单温度探针自相关函数和算子Krylov结构，无需对称性、参数模型或受控操作，即可重建未知系统的谱信息，并具有多项式样本复杂度保证。
AI 中文摘要
自然量子系统的哈密顿量通常是通过间接方式确定的，即提出微观模型并将其预测的观测量与实验进行比较。更直接的方法是从系统本身的测量中提取关于哈密顿量的信息。然而，在许多情况下，无法直接访问整个系统，只能通过可控探针进行测量，如量子传感和边界光谱学。在这种受限访问下，现有的哈密顿量学习协议通常需要特殊的对称性、已知的参数形式，或超出自由演化的受控操作。我们引入了量子Krylov学习（QKL），一种避免这些要求的受限访问框架。QKL利用单温度探针的自相关函数及其算子Krylov结构，提取未知系统的模型无关谱信息。它不是拟合参数化的哈密顿量，而是重建表示探针可访问的动态模式的算子Krylov Jacobi矩阵。该方法假设未观测系统处于或接近无限温度。在此条件下，Jacobi矩阵可以直接从探针可观测量的自相关函数重建，无需对哈密顿量的预言机访问。所需操作是在未知哈密顿量$H$下的自由演化，以及探针态制备和测量。因此，学习核心不需要对称性、参数模型或受控酉门。我们展示了算子Krylov链的Lieb-Robinson界，并推导了所需观测窗口的数据驱动判据。我们还建立了端到端的样本复杂度保证：在谱间隙和权重的通用条件下，达到谱精度$\varepsilon$所需的测量次数是$1/\varepsilon$的多项式。
英文摘要
The Hamiltonian of a natural quantum system is usually determined indirectly, by proposing microscopic models and comparing their predicted observables with experiment. A more direct approach is to extract information about the Hamiltonian from measurements on the system itself. In many settings, however, direct access to the full system is unavailable and measurements can be performed only through controllable probes, as in quantum sensing and boundary spectroscopy. Existing Hamiltonian learning protocols under such restricted access typically require a special symmetry, a known parametric form, or controlled operations beyond free evolution. We introduce Quantum Krylov Learning (QKL), a restricted-access framework that avoids these requirements. QKL uses single temperature probe autocorrelators and their operator Krylov structure to extract model-independent spectral information about the unknown system. Rather than fitting a parametrized Hamiltonian, it reconstructs the operator Krylov Jacobi matrix representing the dynamical modes accessible to the probe. The method assumes that the unobserved system is at, or near, infinite temperature. Under this condition, the Jacobi matrix can be reconstructed directly from the autocorrelation of a probe observable, without oracle access to the Hamiltonian. The required operations are free evolution under the unknown $H$, together with probe state preparation and measurement. The learning core therefore requires no symmetry, no parametric model, and no controlled-unitary gates. We show a Lieb--Robinson bound for the operator Krylov chain and derive a data-driven criterion for the required observation window. We also establish an end-to-end sample complexity guarantee: under generic conditions on the spectral gaps and weights, the number of measurement shots required to achieve spectral precision $\varepsilon$ is polynomial in $1/\varepsilon$.
Comments46 pages, 7 figures
