  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 single-particle
unitaries leads to an identification of the average mixing rate as being
determined solely by the entangling power $e_p(U)$. The same considerations
extend to disordered dual-unitary circuits as well. 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. These constructions and results pave the way for a systematic
extension of quantities studied in the dual-unitary circuits as well as provide
natural generalizations to the non-dual cases.
