Shivan Mittal

We consider random quantum circuits (RQC) on arbitrary connected graphs whose edges determine the allowed 2-qudit interactions. Prior work has established that such n-qudit circuits with local dimension q on 1D, complete, and D-dimensional graphs form approximate unitary designs, that is, they generate unitaries from distributions close to the Haar measure on the unitary group U(q^n) after polynomially many gates. Here, we extend those results by proving that RQCs comprised of O(poly(n,k)) gates on a wide class of graphs form approximate unitary k-designs. We prove that RQCs on graphs with spanning trees of bounded degree and height form k-designs after O(|E|n rm poly(k)) gates, where |E| is the number of edges in the graph. Furthermore, we identify larger classes of graphs for which RQCs generate approximate designs in polynomial circuit size. For k łeq 4, we show that RQCs on graphs of certain maximum degrees form designs after O(|E|n) gates, providing explicit constants. We determine our circuit size bounds from the spectral gaps of local Hamiltonians. To that end, we extend the finite-size (or Knabe) method for bounding gaps of frustration-free Hamiltonians on regular graphs to arbitrary connected graphs. We further introduce a new method based on the Detectability Lemma for determining the spectral gaps of Hamiltonians on arbitrary graphs. Our methods have wider applicability as the first method provides a succinct alternative proof of [Commun. Math. Phys. 291, 257 (2009)] and the second method proves that RQCs on any connected architecture form approximate designs in quasi-polynomial circuit size.