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 kc-design with kc=O(Ld) for a system in d spatial dimensions with linear dimension L. Second, for k<kc, we derive bounds on the depth τ required for the circuit to converge to an approximate k-design. The depth is lower bounded by diffusion kL2ln(L)≲τ. In contrast, without number conservation τ∼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 τ≲kLd+2. Published by the American Physical Society 2025