Uniform Mixing and Association Schemes

TL;DR

This paper investigates conditions under which continuous-time quantum walks on distance-regular graphs exhibit uniform mixing, identifying specific graph classes where this phenomenon occurs or is excluded, using algebraic and combinatorial methods.

Contribution

The paper characterizes graphs with uniform mixing using association schemes and complex Hadamard matrices, providing new constraints and examples for quantum walk uniform mixing.

Findings

Only certain strongly regular graphs admit instantaneous uniform mixing.

Uniform mixing on bipartite graphs requires the number of vertices to be divisible by four.

Uniform mixing does not occur on cycles C_{2m} for m >= 3 or on prime cycles C_p with p >= 5.

Abstract

We consider continuous-time quantum walks on distance-regular graphs of small diameter. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing. First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph X with n vertices, then n is divisible by four. We also prove that if X is bipartite and regular, then n is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on C_{2m} for m >= 3. Using a result of Haagerup, we show that uniform mixing does not occur on C_p for any prime p such that p >= 5. In contrast to this result, we see that epsilon-uniform mixing occurs on C_p for all primes p.