Simulating certain aspects of many-body quantum dynamics with random Clifford circuits

Unitary circuit models provide a useful lens on the universal aspects of the dynamics of many-body quantum systems. Here, we consider random circuits composed of Clifford gates which enable simulations of large system sizes even on classical computers. We study different circuit geometries, including one-dimensional random circuits with long-range gates, as well as Floquet circuits in one and two dimension where short-ranged gates are spatially random but periodic in time. We show that random Clifford circuits can faithfully capture the interplay of hydrodynamic transport and entanglement growth, where the buildup of entanglement is constrained by the presence of a conservation law in the system. Furthermore, by simulating the spreading of local operators, we demonstrate that Clifford circuits can exhibit signatures of both, localization and ergodicity, depending on the circuit geometry. In this context, we also study the spectral form factor of the Floquet Clifford unitary and unveil that it exhibits an exponential ramp, similar to quasi-free fermions with chaotic single-particle dynamics. Our work explores the possibility of simulating exotic nonequilibrium quantum phenomena using Clifford circuits and elucidates the differences and similarities between Clifford dynamics and more generic types of quantum systems.