# Universality in perfect state transfer

## TL;DR

This paper characterizes graphs with universal perfect state transfer in quantum walks, extends previous results, constructs new non-circulant examples, and establishes conditions under which circulants must be complete for universal transfer.

## Contribution

It provides new characterizations of graphs with universal perfect state transfer, constructs non-circulant examples, and proves conditions for circulants to be complete.

## Key findings

- New characterizations of graphs with universal perfect state transfer.
- Construction of non-circulant graphs with universal perfect state transfer.
- Circulant graphs with prime power order must be complete for universal transfer.

## Abstract

A continuous-time quantum walk on a graph is a matrix-valued function $\exp(-\mathtt{i} At)$ over the reals, where $A$ is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices $u,v$, there is a time where the $(v,u)$ entry of the matrix exponential has unit magnitude. We prove new characterizations of graphs with universal perfect state transfer. This extends results of Cameron et al. (Linear Algebra and Its Applications, 455:115-142, 2014). Also, we construct non-circulant families of graphs with universal perfect state transfer. All prior known constructions were circulants. Moreover, we prove that if a circulant, whose order is prime, prime squared, or a power of two, has universal perfect state transfer then its underlying graph must be complete. This is nearly tight since there are universal perfect state transfer circulants with non-prime-power order where some edges are missing.