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.
Highlights
The complexity of interacting many-body systems, classical or quantum, matches their importance and prevalence that one cannot overstate
The exponential fast mixing subsumes the case of K systems. While this quantum classification has not singled out an equivalent of Bernoulli systems, we show that when the local systems are larger than two-dimensional, that is qutrits or above, it is possible for the dual-unitary circuit to become Bernoulli in the sense that CAB(t ; x, y) = C0δt,0δx,y, it vanishes at all space-time separated points
One may first perform the average on the spectral radius |λ1| before finding the rate. All of these quantities, measuring the mixing induced by the channel, are by construction local unitary invariance (LUI) and we address their connection to LUI measures based on U itself, in particular to the entangling power ep(U )
Summary
The complexity of interacting many-body systems, classical or quantum, matches their importance and prevalence that one cannot overstate. Random unitary circuit models involving random twobody local interactions have been extensively studied in the recent past [19,20,21,22,23] as minimal models of many-body quantum nonintegrable systems. The key aspect that make these models solvable is an underlying space-time duality that is realized for certain interactions, these are referred to as dual-unitary circuits [26] Quantities such as the two-point correlation functions, operator entanglements, spectral form factors, have become largely analytically accessible in these models [26,27,29,30,31,32,33,34,35,36]. Computational power of the circuits built from one and twodimensional dual-unitary operators are studied recently and shows the quantum computational supremacy of these circuits [41]
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