Stochastic Sampling of Operator Growth Dynamics

We put forward a Monte Carlo algorithm that samples the Euclidean time operator growth dynamics at infinite temperature. Crucially, our approach is free from the numerical sign problem for a broad family of quantum many-body spin systems, allowing for numerically exact and unbiased calculations. We apply this methodological headway to study the high-frequency dynamics of the mixed-field quantum Ising model (QIM) in one and two dimensions. The resulting quantum dynamics display rapid thermalization, supporting the recently proposed operator growth hypothesis. Physically, our findings correspond to an exponential fall-off of generic response functions of local correlators at large frequencies. Remarkably, our calculations are sufficiently sensitive to detect subtle logarithmic corrections of the hypothesis in one dimension. In addition, in two dimensions, we uncover a non-trivial dynamical crossover between two large frequency decay rates. Lastly, we reveal spatio-temporal scaling laws associated with operator growth, which are found to be strongly affected by boundary contributions.