Sumner Hearth, Boston University

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 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. For k < kc, the depth τ to converge to a k-design scales as τ \gtrsim k Ld+2.  In contrast, without number conservation τ \gtrsim k Ld. 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. The associated Goldstone modes are gapless and lead to the predicted scaling of τ.  Our variational bounds hold for arbitrary spatial and qudit dimensions; we conjecture they are tight.