# Extremal jumps of circuit complexity of unitary evolutions generated by random Hamiltonians

arXiv (Cornell University)（2023）

Abstract

We investigate circuit complexity of unitaries generated by time evolution of
randomly chosen strongly interacting Hamiltonians in finite dimensional Hilbert
spaces. Specifically, we focus on two ensembles of random generators – the so
called Gaussian Unitary Ensemble (GUE) and the ensemble of diagonal Gaussian
matrices conjugated by Haar random unitary transformations. In both scenarios
we prove that the complexity of exp(-it H) exhibits a surprising behaviour
– with high probability it reaches the maximal allowed value on the same time
scale as needed to escape the neighborhood of the identity consisting of
unitaries with trivial (zero) complexity. We furthermore observe similar
behaviour for quantum states originating from time evolutions generated by
above ensembles and for diagonal unitaries generated from the ensemble of
diagonal Gaussian Hamiltonians. To establish these results we rely heavily on
structural properties of the above ensembles (such as unitary invariance) and
concentration of measure techniques. This gives us a much finer control over
the time evolution of complexity compared to techniques previously employed in
this context: high-degree moments and frame potentials.

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Key words

unitary evolutions,circuit complexity,extremal jumps

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