Unitary k-designs from random number-conserving quantum circuits

Local random circuits scramble efficiently and accordingly have a range of applications in quantum information and quantum dynamics. With a global \(U(1)\) charge however, the scrambling ability is reduced; for example, such random circuits do not generate the entire group of number-conserving unitaries. We establish two results using the statistical mechanics of \(k\)-fold replicated circuits. First, we show that finite moments cannot distinguish the ensemble that local random circuits generate from the Haar ensemble on the entire group of number-conserving unitaries. Specifically, the circuits form a \(k_c\)-design with \(k_c = O(L^d)\) for a system in \(d\) spatial dimensions with linear dimension \(L\). Second, for \(k < k_c\), we derive bounds on the depth \(\tau\) required for the circuit to converge to an approximate \(k\)-design. The depth is lower bounded by diffusion \(k L^2 \ln(L) \lesssim \tau\). In contrast, without number conservation \(\tau \sim \text{poly}(k) L\). The convergence of the circuit ensemble is controlled by the low-energy properties of a frustration-free quantum statistical model which spontaneously breaks \(k\) \(U(1)\) symmetries. We conjecture that the associated Goldstone modes set the spectral gap for arbitrary spatial and qudit dimensions, leading to an upper bound \(\tau \lesssim k L^{d+2}\).