20 July 2021 Tue 4 pm

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We study the time evolution of a simple operator via its journey along the Krylov basis: an ordered basis particularly suited for the operator's time evolution.  The associated Lanczos-sequence encodes the features of the operator's journey along its Krylov basis.  Krylov complexity (K-complexity) of the operator is defined to be its average location on this (ordered) basis, as a function of time.  

We find an upper bound for the dimension of the Krylov subspace, and argue that this bound will be saturated for a simple operator in any quantum many-body chaotic system.  We thus provide a natural bound for K-complexity in finite systems.

Complete numerical results for SYK_4 will be presented, in particular the full Lanczos-sequence and K-complexity for all time scales, which were computed using large computer clusters.

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