Unitary k-designs from random number-conserving quantum circuits

In a number conserving quantum system the kth-Renyi entropy can be extracted from statistical correlations between measurements in random bases consistent with the conservation law. Unlike systems without a conserved number, the entropy cannot be extracted from the measurements of circuits composed of spatially non-overlapping unitaries. Instead, the minimal circuit has a brick layer structure with depth scaling with the square of the linear dimension for a maximal error in the purity of order one. This scaling of the depth is a direct a consequence of the diffusive dynamics of the conserved number. Surprisingly, we observe this scaling even when the number density in the state is uniform, e.g. in states with hyperuniform particle distributions and uniformly delocalized particles.