From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy

Abstract

Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are ``as random as a coin toss.'' Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power ${e}_{p}(U)$ of the basic two-particle unitary building block, $U$, of the circuit that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally, we show, both analytically and numerically, how local averaging over random realizations of the single-particle unitaries ${u}_{i}$ and ${v}_{i}$ such that the building block is ${U}^{\ensuremath{'}}=({u}_{1}\ensuremath{\bigotimes}{u}_{2})U({v}_{1}\ensuremath{\bigotimes}{v}_{2})$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power ${e}_{p}(U)$. Finally, we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map, which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.