Unitary k -Designs from Random Number-Conserving Quantum Circuits
Sumner N. Hearth, Michael O. Flynn, Anushya Chandran, Chris R. Laumann · Physical Review X · 2025
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 τ required for the circuit to converge to an approximate k -design. The depth is lower bounded by diffusion k L 2 ln ( 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 ≲ k L d + 2 .