Designs from Local Random Quantum Circuits with $SU(d)$ Symmetry

The generation of $k$-designs (pseudorandom distributions that emulate the Haar measure up to $k$ moments) with local quantum circuit ensembles is a problem of fundamental importance in quantum information and physics. Despite the extensive understanding of this problem for ordinary random circuits, the crucial situations in which symmetries or conservation laws are in play are known to pose fundamental challenges and remain little understood. Here, we construct explicit local unitary ensembles that can achieve high-order unitary $k$-designs under transversal continuous symmetry, in the particularly important $SU⁡(d)$ case. Specifically, we define the convolutional quantum alternating (CQA) group generated by 4-local $SU⁡(d)$-symmetric Hamiltonians as well as associated 4-local $SU⁡(d)$-symmetric random unitary circuit ensembles and prove that they form and converge to $SU⁡(d)$-symmetric $k$-designs, respectively, for all $k < n(n − 3)/2$, with $n$ being the number of qudits. A key technique that we employ to obtain the results is the Okounkov-Vershik approach to $S_n$ representation theory. To study the convergence time of the CQA ensemble, we develop a numerical method using the Young orthogonal form and the $S_n$ branching rule. We provide strong evidence for a subconstant spectral gap and certain convergence time scales of various important circuit architectures, which contrast with the symmetry-free case. We also provide comprehensive explanations of the difficulties and limitations in rigorously analyzing the convergence time using methods that have been effective for cases without symmetries, including Knabe's local gap threshold and Nachtergaele's martingale methods. This suggests that a novel approach is likely necessary for understanding the convergence time of $SU(d)$-symmetric local random circuits.