Perfect state transfer in products and covers of graphs

TL;DR

This paper investigates which graphs allow perfect quantum state transfer, focusing on tensor product graphs, and constructs new examples demonstrating this phenomenon in quantum information networks.

Contribution

It introduces new methods to identify perfect state transfer in tensor product graphs, expanding the class of graphs known to support this quantum property.

Findings

Constructed new examples of graphs with perfect state transfer

Analyzed tensor product graphs for quantum state transfer

Provided conditions under which perfect transfer occurs

Abstract

A continuous-time quantum walk on a graph $X$ is represented by the complex matrix $\exp (-\mathrm{i} t A)$, where $A$ is the adjacency matrix of $X$ and $t$ is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of $01$-matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.