Abstract Recently introduced dual unitary brickwork circuits have been recognised as paradigmatic exactly solvable quantum chaotic many-body systems with tunable degree of ergodicity and mixing. Here we show that regularity of the circuit lattice is not crucial for exact solvability. We consider a circuit where random 2-qubit dual unitary gates sit at intersections of random arrangements of straight lines in two dimensions (mikado) and analytically compute the variance of the spatio-temporal correlation function of local operators. Note that the average correlator vanishes due to local Haar randomness of the gates. The result can be physically motivated for two random mikado settings. The first corresponds to the thermal state of free particles carrying internal qubit degrees of freedom which experience interaction at kinematic crossings, while the second represents rotationally symmetric (random euclidean) space-time.
Abstract Recently introduced dual unitary brickwork circuits have been recognised as paradigmatic exactly solvable quantum chaotic many-body systems with tunable degree of ergodicity and mixing. Here we show that regularity of the circuit lattice is not crucial for exact solvability. We consider a circuit where random 2-qubit dual unitary gates sit at intersections of random arrangements of straight lines in two dimensions (mikado) and analytically compute the variance of the spatio-temporal correlation function of local operators. Note that the average correlator vanishes due to local Haar randomness of the gates. The result can be physically motivated for two random mikado settings. The first corresponds to the thermal state of free particles carrying internal qubit degrees of freedom which experience interaction at kinematic crossings, while the second represents rotationally symmetric (random euclidean) space-time.
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 ep(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 ui and vi such that the building block is ${U}^{\ensuremath{'}}=({u}_{1}\ensuremath{\bigotimes}{u}_{2})U({v}_{1}\ensuremath{\bigotimes}{v}_{2})$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power ep(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.
Abstract Dual-unitary brickwork circuits are an exactly-solvable model for many-body chaotic quantum systems, based on 2-site gates which are unitary in both the time and space directions. Prosen has recently described an alternative model called dual-unitary interactions round-a-face, which we here call clockwork, based on 2-controlled 1-site unitaries composed in a non-brickwork structure, yet with many of the same attractive global properties. We present a 2-categorical framework that simultaneously generalizes these two existing models, and use it to show that brickwork and clockwork circuits can interact richly, yielding new types of generalized heterogeneous circuits. We show that these interactions are governed by quantum combinatorial data, which we precisely characterize. These generalized circuits remain exactly-solvable and we show that they retain the attractive features of the original models such as single-site correlation functions vanishing everywhere except on the causal light-cone. Our framework allows us to directly extend the notion of solvable initial states to these biunitary circuits, and we show these circuits demonstrate maximal entanglement growth and exact thermalization after finitely many time steps.
We consider one-dimensional quantum circuits of the brickwork type, where the fundamental quantum gate is dual unitary. Such models are solvable: the dynamical correlation functions of the infinite temperature ensemble can be computed exactly. We review various existing constructions for dual unitary gates and we supplement them with new ideas in a number of cases. We discuss connections with various topics in physics and mathematics, including quantum information theory, tensor networks for the AdS/CFT correspondence (holographic error correcting codes), classical combinatorial designs (orthogonal Latin squares), planar algebras, and Yang-Baxter maps. Afterwards we consider the ergodicity properties of a special class of dual unitary models, where the local gate is a permutation matrix. We find an unexpected phenomenon: nonergodic behavior can manifest itself in multisite correlations, even in those cases when the one-site correlation functions are fully chaotic (completely thermalizing). We also discuss the circuits built out of perfect tensors. They appear locally as the most chaotic and most scrambling circuits, nevertheless they can show global signs of nonergodicity: if the perfect tensor is constructed from a linear map over finite fields, then the resulting circuit can show exact quantum revivals at unexpectedly short times. A brief mathematical treatment of the recurrence time in such models is presented in Appendix (written by Roland Bacher and Denis Serre).
We study the scrambling of quantum information in local random unitary circuits by focusing on the tripartite information proposed by Hosur et al. We provide exact results for the averaged R\'enyi-2 tripartite information in two cases: (i) the local gates are Haar random and (ii) the local gates are dual-unitary and randomly sampled from a single-site Haar-invariant measure. We show that the latter case defines a one-parameter family of circuits, and prove that for a ``maximally chaotic'' subset of this family quantum information is scrambled faster than in the Haar-random case. Our approach is based on a standard mapping onto an averaged folded tensor network, that can be studied by means of appropriate recurrence relations. By means of the same method, we also revisit the computation of out-of-time-ordered correlation functions, rederiving known formulas for Haar-random unitary circuits, and presenting an exact result for maximally chaotic random dual-unitary gates.
Dual-unitary circuits are paradigmatic examples of exactly solvable yet chaotic quantum many-body systems, but solvability naturally goes along with a degree of nongeneric behavior. By investigating the effect of weakly broken dual unitarity on the spreading of local operators, we study whether, and how, small deviations from dual unitarity recover fully generic many-body dynamics. We present a discrete path-integral formula for the out-of-time-order correlator and recover a butterfly velocity smaller than the light-cone velocity, v_{B}<v_{LC}, and a diffusively broadening operator front, two generic features of ergodic quantum spin chains absent in dual-unitary circuit dynamics. The butterfly velocity and diffusion constant are determined by a small set of microscopic quantities, and the operator entanglement of the gates has a crucial role.
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.