Unitary k-designs from random number-conserving quantum circuits

Abstract

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 $\tau \lesssim k{L}^{d+2}$.

18 pages, 2 figures

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