Krylov Break Times from an Inhomogeneous Lieb--Robinson Light Cone

Le résumé fourni par la source

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

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