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We consider random quantum circuits (RQC) on arbitrary connected graphs whoseedges determine the allowed $2$-qudit interactions. Prior work has establishedthat such $n$-qudit circuits with local dimension $q$ on 1D, complete, and$D$-dimensional graphs form approximate unitary designs, that is, they generateunitaries from distributions close to the Haar measure on the unitary group$U(q^n)$ after polynomially many gates. Here, we extend those results byproving that RQCs comprised of $O(\mathrm{poly}(n,k))$ gates on a wide class ofgraphs form approximate unitary $k$-designs. We prove that RQCs on graphs withspanning trees of bounded degree and height form $k$-designs after$O(|E|n\,\mathrm{poly}(k))$ gates, where $|E|$ is the number of edges in thegraph. Furthermore, we identify larger classes of graphs for which RQCsgenerate approximate designs in polynomial circuit size. For $k \leq 4$, weshow that RQCs on graphs of certain maximum degrees form designs after$O(|E|n)$ gates, providing explicit constants. We determine our circuit sizebounds from the spectral gaps of local Hamiltonians. To that end, we extend thefinite-size (or Knabe) method for bounding gaps of frustration-freeHamiltonians on regular graphs to arbitrary connected graphs. We furtherintroduce a new method based on the Detectability Lemma for determining thespectral gaps of Hamiltonians on arbitrary graphs. Our methods have widerapplicability as the first method provides a succinct alternative proof of[Commun. Math. Phys. 291, 257 (2009)] and the second method proves that RQCs onany connected architecture form approximate designs in quasi-polynomial circuitsize.

Local random quantum circuits form approximate designs on arbitrary architectures