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

S. Aravinda; Suhail Ahmad Rather; Arul Lakshminarayan

doi:10.1103/PhysRevResearch.3.043034

paper

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

arXiv:2101.04580 · doi:10.1103/PhysRevResearch.3.043034

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 $U^{'} = (u_{1} \otimes u_{2}) U (v_{1} \otimes v_{2})$ 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.

arXiv comment48 pages (38 pages main text + 10 pages Appendix ), 16 (10+6) figures. Contains new results and substantially modified

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

Abstract

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