On Information Propagation in Krylov Subspaces

Abstract:Krylov subspace methods provide an efficient tool for modeling any quantum initial value problem as a one dimensional chain. A time ordered path formulation, in which the dynamical generator successively splits probability between paths, yields a speed limit within such dynamical subspaces. For power law growing Lanczos coefficients $b_n \sim n^\delta$ we find super ballistic lightcones, with an exponentially spreading lightcone for maximal growth $\delta = 1$. Causality in the Krylov chain constrains the Lanczos coefficients to grow at most linearly. We illustrate the validity of the above in the $SL(2, R)$ and Heisenberg Weyl algebras, and in a many body skew field Ising model. Moreover, we extend the above bound to the family of $Q$ complexities. Lastly, mapping to a spin chain gives a Lieb Robinson bound analogue and opens the problem to standard condensed matter methods.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th)

arXiv:2610.03883 [quant-ph]

(or arXiv:2610.03883v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.03883 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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