Local random quantum circuits form approximate designs on arbitrary architectures

• URL: http://arxiv.org/abs/2310.19355v1
• Date: Mon, 30 Oct 2023 08:48:14 GMT
• Title: Local random quantum circuits form approximate designs on arbitrary architectures
• Authors: Shivan Mittal, Nicholas Hunter-Jones
• Abstract summary: We consider random quantum circuits (RQC) on arbitrary connected graphs whose edges determine the allowed $2$-qudit interactions. We prove that RQCs on graphs with spanning trees of bounded degree and height form $k$-designs after $O(mathrmpoly(k))$ gates. We identify larger classes of graphs for which RQCs generate approximate designs in circuit size.
• Score: 0.1977825609388727
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $O(\mathrm{poly}(n,k))$ 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 $O(|E|n\,\mathrm{poly}(k))$ 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 $k \leq 4$, 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.

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