Complex Hadamard Matrices, Instantaneous Uniform Mixing and Cubes

TL;DR

This paper investigates quantum walks on hypercube-related graphs, identifying conditions for uniform mixing and perfect state transfer, and characterizing graphs with complex Hadamard matrices in their adjacency algebra.

Contribution

It introduces new classes of graphs with uniform mixing and perfect state transfer properties, and characterizes those containing complex Hadamard matrices.

Findings

Graphs in the adjacency algebra of certain hypercubes admit instantaneous uniform mixing.

Identifies graphs with perfect state transfer at specific times.

Characterizes folded and halved cubes with complex Hadamard matrices.

Abstract

We study the continuous-time quantum walks on graphs in the adjacency algebra of the $n$-cube and its related distance regular graphs. For $k\geq 2$, we find graphs in the adjacency algebra of $(2^{k+2}-8)$-cube that admit instantaneous uniform mixing at time $\pi/2^k$ and graphs that have perfect state transfer at time $\pi/2^k$. We characterize the folded $n$-cubes, the halved $n$-cubes and the folded halved $n$-cubes whose adjacency algebra contains a complex Hadamard matrix. We obtain the same conditions for the characterization of these graphs admitting instantaneous uniform mixing.