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Local random circuits scramble efficiently and accordingly have a range ofapplications in quantum information and quantum dynamics. With a global $U(1)$charge however, the scrambling ability is reduced; for example, such randomcircuits do not generate the entire group of number-conserving unitaries. Weestablish two results using the statistical mechanics of $k$-fold replicatedcircuits. First, we show that finite moments cannot distinguish the ensemblethat local random circuits generate from the Haar ensemble on the entire groupof number conserving unitaries. Specifically, the circuits form a $k_c$-designwith $k_c = O(L^d)$ for a system in $d$ spatial dimensions with lineardimension $L$. Second, for $k < k_c$, the depth $\tau$ to converge to a$k$-design scales as $\tau \gtrsim k L^{d+2}$. In contrast, without numberconservation $\tau \gtrsim k L^{d}$. The convergence of the circuit ensemble iscontrolled by the low-energy properties of a frustration-free quantumstatistical model which spontaneously breaks $k$ $U(1)$ symmetries. Theassociated Goldstone modes are gapless and lead to the predicted scaling of$\tau$. Our variational bounds hold for arbitrary spatial and qudit dimensions;we conjecture they are tight.

Unitary k-designs from random number-conserving quantum circuits