Entanglement Growth and Minimal Membranes in (d+1) Random Unitary Circuits

Understanding the nature of entanglement growth in many-body systems is one of the fundamental questions in quantum physics. Here, we study this problem by characterizing the entanglement fluctuations and distribution of a (d+1)-dimensional qubit lattice evolved under a random unitary circuit. Focusing on Clifford gates, we perform extensive numerical simulations of random circuits in 1≤d≤4 dimensions. Our findings demonstrate that properties of growth of bipartite entanglement entropy are characterized by the roughening exponents of a d-dimensional membrane in a (d+1)-dimensional elastic medium.