Mixing of quantum walk on circulant bunkbeds (pp370-381)
         Peter Lo, Siddharth Rajaram, Diana Schepens, Daniel Sullivan, Chritino Tamon, and Jeffrey Ward
          doi: https://doi.org/10.26421/QIC6.4-5-5
Abstracts: This paper gives new observations on the mixing dynamics of a continuous-time quantum walk on circulants and their bunkbeds. These bunkbeds are defined through two standard graph operators: the join G + H and the Cartesian product G \cprod H of graphs G and H. Our results include the following:  (i) The quantum walk is average uniform mixing on circulants with bounded eigenvalue multiplicity; this extends a known fact about the cycles C_{n}.  (ii) Explicit analysis of the probability distribution of the quantum walk on the join of circulants; this explains why complete multipartite graphs are not average uniform mixing, using the fact K_{n} = K_{1} + K_{n-1} and K_{n,\ldots,n} = \overline{K}_{n} + \ldots + \overline{K}_{n}.  (iii) The quantum walk on the Cartesian product of a $m$-vertex path P_{m} and a circulant G, namely, P_{m} \cprod G, is average uniform mixing if G is; this highlights a difference between circulants and the hypercubes Q_{n} = P_{2} \cprod Q_{n-1}. Our proofs employ purely elementary arguments based on the spectra of the graphs.
Key words: quantum walks, continuous-time, mixing, circulants