  We put forward a Monte Carlo algorithm which samples the Euclidean time
operator growth dynamics at infinite temperature. Crucially, our approach is
free of the numerical sign problem for a range 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) at one and two dimensions. Our results
corroborate the recently proposed operator growth hypothesis, predicting an
exponential fall-off of generic response functions 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 crossover between two large frequency
decay rates. Lastly, we find that the growth rate of the operator support size
is dominated by boundary effects, with a logarithmic correction in one
dimension.
