Uniform Mixing on Cayley Graphs

Abstract

We provide new examples of Cayley graphs on which the quantum walks reach uniform mixing. Our first result is a complete characterization of all $2(d+2)$-regular Cayley graphs over ${\mathbb{Z}}_3^d$ that admit uniform mixing at time $2\pi /9$. Our second result shows that for every integer $k\ge 3$, we can construct Cayley graphs over ${\mathbb{Z}}_q^d$ that admit uniform mixing at time $2\pi /q^k$, where $q=3,4$. We also find the first family of irregular graphs, the Cartesian powers of the star $K_{1,3}$, that admit uniform mixing.

References in corpus (4)

• Complex Hadamard Matrices, Instantaneous Uniform Mixing and Cubes
• Generalized Hamming schemes
• Mixing in Continuous Quantum Walks on Graphs
• Universal Mixing of Quantum Walk on Graphs