Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs

Abstract

The quadratic unitary Cayley graph ${\mathcal{G}}_{{\mathbb{Z}}_n}$ has vertex set ${\mathbb{Z}}_n:=\{0,1,\dots ,n-1\}$, where two vertices $u$ and $v$ are adjacent if and only if $u-v$ or $v-u$ is a square of some units in ${\mathbb{Z}}_n$. This paper explores the periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs. We determine all periodic quadratic unitary Cayley graphs. From our results, it follows that there are infinitely many integral as well as non-integral graphs that are periodic. Additionally, we also determine the values of $n$ for which the quadratic unitary Cayley graph ${\mathcal{G}}_{{\mathbb{Z}}_n}$ exhibits perfect state transfer.

References in corpus (5)

• Perfect state transfer by means of discrete-time quantum walk search algorithms on highly symmetric graphs
• Periodicity of quantum walks defined by mixed paths and mixed cycles
• Quadratic unitary Cayley graphs of finite commutative rings
• Perfect state transfer on hypercubes and its implementation using superconducting qubits
• Combinatorial necessary conditions for regular graphs to induce periodic quantum walks