Many-body quantum chaos and dual-unitarity round-a-face

We propose a new type of locally interacting quantum circuits—quantum cellular automata—that are generated by unitary interactions round-a-face (IRF). Specifically, we discuss a set (or manifold) of dual-unitary IRFs with local Hilbert space dimension $d$ [DUIRF$(d)$], which generate unitary evolutions both in space and time directions of an extended 1+1 dimensional lattice. We show how arbitrary dynamical correlation functions of local observables can be evaluated in terms of finite-dimensional completely positive trace preserving unital maps in complete analogy to recently studied circuits made of dual-unitary brick gates (DUBGs). The simplest non-vanishing local correlation functions in dual-unitary IRF circuits are shown to involve observables non-trivially supported on two neighboring sites. We completely characterize the ten-dimensional manifold of DUIRF$(2)$ for qubits (⁠$d=2$⁠) and provide, for $d=3,4,…,7$⁠, empirical estimates of its dimensionality based on numerically determined dimensions of tangent spaces at an ensemble of random instances of dual-unitary IRF gates. In parallel, we apply the same algorithm to determine $dimDUBG(d)$ and show that they are of similar order though systematically larger than $dimDUIRF(d)$ for $d=2,3,…,7$⁠. It is remarkable that both sets have a rather complex topology for $d≥3$ in the sense that the dimension of the tangent space varies among different randomly generated points of the set. Finally, we provide additional data on dimensionality of the chiral extension of DUBG circuits with distinct local Hilbert spaces of dimensions $d≠d′$ residing at even/odd lattice sites.