非均匀Lieb-Robinson光锥下的Krylov中断时间
Krylov Break Times from an Inhomogeneous Lieb--Robinson Light Cone
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中文总结 AI 辅助
该研究针对量子动力学等领域的Krylov近似，推导了非均匀输运度量下的Lieb-Robinson界，得到Krylov截断的中断时间下界，输运受限 regime 下中断时间近似等于对应求和，数值测试验证了该结论。
中文摘要 AI 辅助
Krylov和Lanczos近似被应用于量子动力学、量子子空间方法与哈密顿学习中，一个实际问题是m维Krylov截断的可信时长。本文认为该时长由关联雅可比链上的因果传播决定，相关距离并非Krylov索引本身，而是非均匀输运度量ρ(m,n)=∑（j从min(m,n)到max(m,n)-1）1/b_j，其中b_j是键j↔j+1上的Lanczos跃迁。我们证明了该度量下的Lieb-Robinson界，其小权重极限给出速度v_LR=2，即传播在锥ρ(m,n)≃2|t|外呈指数抑制。有限Krylov近似对返回振幅的误差是往返效应：信息需从探针传播到截断边界再返回。结合Lieb-Robinson界与Duhamel公式，可得到截断动力学误差的下界。对于固定容差ε，中断时间t_*(m;ε)指m维截断保证以误差ε重现精确返回振幅的最长时间，我们证明t_*(m;ε)≥τ_m[1-o(1)]，其中τ_m=ρ(0,m)=∑（j<m）1/b_j。当探针沿链传播时，该下界是紧的，故t_*(m)≃τ_m；当探针仅激发谱的局域部分或截断已解析的部分时，情况不同，此时基本无信号到达边界，在与状态相关的Krylov维度m_*之外，近似可始终保持准确，中断时间有效为无穷大。对自旋链与随机雅可比矩阵的数值测试支持输运受限 regime 下t_*(m)≃τ_m。
英文摘要
Krylov and Lanczos approximations are used in quantum dynamics, quantum subspace methods, and Hamiltonian learning. A practical question is how long an $m$-dimensional Krylov truncation can be trusted. We argue that this time is fixed by causal propagation on the associated Jacobi chain. The relevant distance is not the Krylov index itself, but the inhomogeneous transport metric $ρ(m,n) = \sum_{j=\min(m,n)}^{\max(m,n)-1} 1/b_j$, where $b_j$ is the Lanczos hopping across the bond $j \leftrightarrow j+1$. We prove a Lieb--Robinson bound in this metric. Its small-weight limit gives the velocity $v_{\rm LR} = 2$, meaning that propagation is exponentially suppressed outside the cone $ρ(m,n) \simeq 2|t|$. The error of a finite Krylov approximation to the return amplitude is a round-trip effect: information has to travel from the probe to the truncation boundary and back. Combining the Lieb--Robinson bound with Duhamel's formula yields a lower bound on the error of the truncated dynamics. For a fixed tolerance $ε$, let the break time $t_\ast(m;ε)$ denote the longest time for which the $m$-dimensional truncation is guaranteed to reproduce the exact return amplitude within error $ε$. We show that $t_\ast(m;ε) \ge τ_m[1-o(1)]$, where $τ_m = ρ(0,m) = \sum_{j<m} 1/b_j$. When the probe spreads along the chain, this lower bound is also tight, so $t_\ast(m) \simeq τ_m$. The situation is different when the probe excites only a localized part of the spectrum, or a part already resolved by the truncation. In this case, essentially no signal reaches the boundary. Beyond a state-dependent Krylov dimension $m_\ast$, the approximation can therefore remain accurate at all times, and the break time is effectively infinite. Numerical tests on spin chains and random Jacobi matrices support $t_\ast(m) \simeq τ_m$ in the transport-limited regime.
