Universality and conformal invariance in hybrid quantum circuits

We establish the emergence of conformal field theories (CFT) in (1+1)-dimensional hybrid quantum circuits right at the measurement-driven entanglement transition, by revealing the remarkable space-time conformal covariance of entanglement entropies and mutual information, computed numerically for various subregions at all time steps within Clifford circuits of up to L = 512 qubits and T = 1024 time steps. Though the evolution takes place in real-time, the time dimension of the circuit becomes “imaginary” or “space-like” – to be consistent with conformal invariance – as a result of measurements that break unitarity. Throughout the paper we investigate Clifford circuits with several different boundary conditions by injecting physical qubits at the spatial and/or temporal boundaries, all giving consistent characterizations of the “Clifford CFT”. We emphasize universal results that are consequences solely of the conformal invariance and do not depend crucially on the precise nature of the CFT. Among these are the infinite “entangling speed” as a manifestation of “many-body entanglement swapping”, and the critical decaying behavior of the thermal entropy of a mixed state at the recently discovered “purification transition”.