Uniform Mixing in Continuous-Time Quantum Walks on Graphs

Benjamin Mustico, Clarkson University

Given a graph G with adjacency matrix A, the continuous-time quantum walk on G is given by the time-varying unitary matrix U(t) = exp(−itA). We say that G has uniform mixing at time t if U(t) is a flat matrix (whose entries have the same absolute value). Ahmadi et al. (2002) showed that a complete graph on n vertices has uniform mixing if and only if n = 2, 3, 4. In this work, we prove that for any complete graph, there is a signing (assignment of complex weights to edges) so that the signed complete graph has uniform mixing. As a further corollary, we show that our signing yields a speedup of uniform mixing time for an infinite family of Hamming graphs. These results illustrate the power of signings (or chirality) for quantum transport on graphs. This is joint work with Luke Levine, Gabe Tucker, Hanmeng Zhan, and Christino Tamon, and is supported by NSF grant OSI-2427020. Note that this is anticipated to be a joint talk with Jessy J. Mesapam.