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

TL;DR

This paper explores the role of entangling power in dual-unitary quantum circuits, establishing conditions for mixing and Bernoulli properties, and provides methods to construct such operators with maximal entangling power.

Contribution

It introduces a condition based on entangling power that guarantees mixing and Bernoulli properties in quantum circuits, extending the ergodic hierarchy to the quantum Bernoulli level.

Findings

Maximized entangling power corresponds to Bernoulli circuits.

Local averaging over random unitaries reveals the dominant role of entangling power in mixing rates.

Constructed dual-unitary operators span the entire range of entangling power, including a coupled quantum cat map.

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 $U^\prime = (u_1 \otimes u_2 ) U (v_1 \otimes v_2 )$ 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.