Majorana Loop Models for Measurement-Only Quantum Circuits

Abstract

Projective measurements in random quantum circuits lead to a rich breadth of entanglement phases and extend the realm of nonunitary quantum dynamics. Here, we explore the connection between measurement-only quantum circuits in one spatial dimension and the statistical mechanics of loop models in two dimensions. While Gaussian Majorana circuits admit a microscopic mapping to loop models, for non-Gaussian, i.e., generic Clifford, circuits, a corresponding mapping may emerge only on a coarse-grained scale. We then focus on a fundamental symmetry of loop models: the orientability of worldlines. We discuss how orientability enters in the measurement framework, acting as a separatrix for the universal long-wavelength behavior in a circuit. When orientability is broken, the circuit falls into the universality class of closely packed loops with crossings (CPLC) and features a Goldstone phase with a peculiar, universal log2(L) scaling of the entanglement entropy. In turn, when orientability is preserved, the long-wavelength behavior of the circuit mimics that of (coupled) two-dimensional Potts models. We demonstrate the strength of the loop model approach by numerically simulating a variety of measurement-only Clifford circuits. Upon varying the set of measured operators, a rich circuit dynamics is observed, ranging from CPLC to the one-state Potts model (percolation), the two-state Potts model (Ising), and coupled Potts models (Berezhinskii-Kosterlitz-Thouless) universality class. Loop models, thus, provide a handle to access a large class of measurement-only circuits and yield a blueprint on how to realize desired entanglement phases by measurement.