Matchgate Circuits Deeply Thermalize.

We study the ensemble of states generated by performing projective measurements on the output of a random matchgate (or free-fermionic) quantum circuit. We rigorously show that this "projected ensemble" exhibits deep thermalization: for large system sizes, it converges toward a universal ensemble that is uniform over the manifold of Gaussian fermionic states. As well as proving momentwise convergence of these ensembles, we demonstrate that the full distribution of any physical observable in the projected ensemble is close to its universal form in Wasserstein-1 distance, which we argue is an appropriate and efficiently computable measure of convergence when studying deep thermalization. Using this metric, we also numerically find that local matchgate circuits deeply thermalize after a timescale t∼L^{2} set by the diffusive spreading of quantum information. Our work opens up new avenues to experimentally accessible protocols to probe the emergence of quantum statistical mechanics and benchmark quantum simulators.