Average Mixing in Chiral Quantum Walks

Jessy Jacob Mesapam, Clarkson University

Given a graph G with adjacency matrix A, the continuous-time quantum walk on G is given by U(t) = exp(−itA). The mixing matrix M(t) of this quantum walk is defined as the Schur product of U(t) with its conjugate. The (a, b)-entry of M(t) represents the probability of observing the quantum walk at b at time t when starting at a. The average mixing matrix (AMM) is the Cesaro limit of M(t) as time t tends to infinity. This matrix encodes the long-term behavior of the quantum walk. We say that a graph has average uniform mixing if its AMM matrix is the normalized all-one matrix. Godsil (2013) proved that no graph has average uniform mixing except for K2. In contrast, we show that there are infinite families of oriented graphs with average uniform mixing. This is joint work with Luke Levine, Benjamin Mustico, Christino Tamon, Gabriel Tucker and Hanmeng Zhan, and is supported by NSF grant OSI-2427020.