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

TL;DR

This paper studies the circuit complexity growth of unitaries generated by random Hamiltonians, revealing that complexity rapidly reaches its maximum in a short time, with implications for quantum dynamics and complexity theory.

Contribution

It demonstrates that the complexity of unitaries from random Hamiltonian evolutions quickly saturates, using novel concentration of measure techniques for better analysis.

Findings

Complexity reaches maximum on the same timescale as escaping trivial unitaries.

Similar complexity behavior observed for quantum states and diagonal unitaries.

Structural properties of ensembles enable finer control over complexity evolution.

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.