Operator Dynamics in Brownian Quantum Circuit

Understand

We explore the operator dynamics in a random $N$-spin model with pairwise interactions (Brownian quanum circuit).

• We introduce the height $h$ of an operator to characterize its spatial extent, and derive the master equation of the height probability distribution.
• The study of an initial simple operator with $h = 1$ (minimal nonzero height) shows that the mean height, which is proportional to the squared commutator, has an initial exponential growth.
• It then slows down around the scrambling time $\sim\log N$ and finally saturates to a steady state in a manner similar to the logistic function.