Classical Models of Entanglement in Monitored Random Circuits

TL;DR

This paper introduces a classical Markov process framework to analyze entanglement dynamics in monitored quantum circuits, linking quantum entropy evolution to classical probabilistic models and percolation theory.

Contribution

It reformulates entanglement evolution in monitored quantum circuits as a classical Markov process and develops algorithms for arbitrary graphs, extending to continuous-time stochastic dynamics.

Findings

Established a probabilistic cellular automaton for entanglement entropy computation.

Linked entropy evolution in 1D to a classical spin model and percolation.

Demonstrated equivalence of zeroth and second Rényi entropy models in large local dimension limit.

Abstract

The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattice percolation problem. We also establish a Markov model for the evolution of the zeroth R\'{e}nyi entropy and demonstrate that, in one dimension and in the limit of large local dimension, it coincides with the corresponding second-R\'{e}nyi-entropy model. Finally, we extend the Markovian description to a more general setting that incorporates continuous-time dynamics, defined by stochastic Hamiltonians and weak local measurements continuously monitoring the system.