Complex Hadamard Matrices, Instantaneous Uniform Mixing and Cubes

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 $k\ge 2$, 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.

24 pages

References in corpus (4)

• Quantum Computation and Decision Trees
• Universal computation by quantum walk
• Perfect state transfer in cubelike graphs
• Generalized Hamming schemes

Cited by in corpus (9)

• Distance-regular graphs
• Perfect state transfer on distance-regular graphs and association schemes
• On state transfer in Cayley graphs for abelian groups
• Perfect State Transfer on Weighted Graphs of the Johnson Scheme
• Discretization of continuous-time quantum walks via the staggered model with Hamiltonians
• Quantum Fractional Revival on Graphs
• A New Perspective on the Average Mixing Matrix
• Uniform Mixing on Cayley Graphs
• Uniform mixing on integral abelian Cayley graph