Random circuit analysis of entanglement measures in noisy, thermalizing quantum dynamics

In Haar random circuits, the exact average over the circuit ensemble results in a 'classical' statistical model with permutation degrees of freedom. For second moment properties, these models are Markovian and have been used as classical descriptions of operator spreading, entanglement growth, and the spreading of errors in noisy circuits. For higher moments, these models are non-Markovian; however, they are still amenable to tensor network methods. We demonstrate key physical properties of the third and fourth moment random circuit statistical models with MPS calculations and as an application analyze Reni entanglement negativity, a quantity that is sensitive to both entanglement growth and noise.