Average mixing of continuous quantum walks

Details Average mixing of continuous quantum walks Average mixing of continuous quantum walks

2011-03-14

arxiv.org/abs/1103.2578v3

Mathematics

Combinatorics

20 pages, minor fixes, added section on discrete walks; fixed typos

Scientific paper

If $X$ is a graph with adjacency matrix $A$, then we define $H(t)$ to be the operator $\exp(itA)$. The Schur (or entrywise) product $H(t)\circ H(-t)$ is a doubly stochastic matrix and, because of work related to quantum computing, we are concerned the \textsl{average mixing matrix}. This can be defined as the limit of $C^{-1} \int_0^C H(t)\circ H(-t)\dt$ as $C\to\infty$. We establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We find that for paths and cycles this matrix takes on a surprisingly simple form, thus for the path it is a linear combination of $I$, $J$ (the all-ones matrix), and a permutation matrix.

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