Speaker: Shivan Mittal (UT Austin)
Title: How does connectivity in random circuits affect convergence to Haar measure?
Abstract: Consider a local random quantum circuit with a connectivity graph on the n-qudits. This is a random walk on the unitary group as follows; pick an edge uniformly at random from the graph and compose a Haar random 2-qudit unitary non-trivially supported on the corresponding qudit Hilbert spaces. This random walk induces a distribution on the unitary group on n-qudits. A circuit is said to form epsilon-approximate unitary k-designs if the k-th moments of the induced distribution are epsilon-close (for specific choices of norm) to the k-th moments of the Haar measure (uniform distribution) on the unitary group. Past literature derived circuit size bounds for convergence of k-th moments on D-dimensional Euclidean lattices like graphs or complete graphs. We extend those circuit size bounds for the case of arbitrary connected graphs via a novel method of lower bounding spectral gaps of Hamiltonians defined on such graphs. Since our work, our method has been improved by other authors (https://arxiv.org/abs/2406.07478) for some connectivity graphs. However, our earlier work and the newer results, still leave open the question, "what property of the connectivity graph is responsible for faster or not convergence?" Time permitting, I will share new results (not on arxiv, but submitted to QIP), where we conjecture a property related to automorphism groups of connectivity graphs that directly affects the circuit size bounds for convergence. In particular, we identify this property in the case of complete graphs and show how that property leads to convergence to epsilon-approximate unitary k-designs in O(n Log(n)) circuit size for k=O(1).
Joint work with Nick Hunter-Jones