Perfect State Transfer in Weighted Cubelike Graphs

TL;DR

This paper extends the concept of perfect state transfer in quantum walks from unweighted cubelike graphs to weighted ones, providing a characterization of weights that enable PST or periodicity at specific times.

Contribution

It generalizes PST and periodicity results to weighted cubelike graphs and characterizes the weights that facilitate these quantum properties.

Findings

Identifies weights that enable PST at t=π/2 in weighted cubelike graphs.

Provides conditions for periodicity in weighted cubelike graphs.

Extends known unweighted graph results to weighted cases.

Abstract

A continuous-time quantum random walk describes the motion of a quantum mechanical particle on an underlying graph. The graph itself is associated with a Hilbert space of dimension equal to the number of vertices. The dynamics of the walk is governed by the unitary operator $\mathcal{U}(t) = e^{iAt}$, where $A$ is the adjacency matrix of the graph. An important notion in the quantum random walk is the transfer of a quantum state from one vertex to another. If the fidelity of the transfer is unity, we call it a perfect state transfer. Many graph families have been shown to admit PST or periodicity, including cubelike graphs. These graphs are unweighted. In this paper, we generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs. We characterize the weights for which they admit PST or show periodicity, both at time $t=\frac{\pi}{2}$.