On Information Propagation in Krylov Subspaces

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 about n δ we find super ballistic lightcones, with an exponentially spreading lightcone for maximal growth δ= 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.