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.
Highlights
A continuous-time quantum walk on a graph X is defined by the transition matrixU(t) := exp(itA) =k k!, k≥0 where A is the adjacency matrix of X
The probability that at time t, the quantum walk with initial state represented by u is in the state represented by v is
We say X admits uniform mixing at time t if the above probability is the same for all vertices u and v
Summary
K≥0 where A is the adjacency matrix of X. We provide new examples of Cayley graphs over Zdq that admit uniform mixing. Theorem 8.5, Theorem 8.6 and Theorem 9.1 show that, for an arbitrarily large integer k, we can construct families of Cayley graphs over Zdq that admit uniform mixing at time 2π/qk, where q = 3, 4. These examples extend the results of Mullin [9] and Chan [3]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
