Quantum Walks on Embeddings

Abstract

We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence between the transition matrix $U$ and the vertex-face incidence structure. Using the incidence graph, we derive a formula for the principal logarithm of $U^2$, and find conditions for its underlying digraph to be an oriented graph. In particular, we show this happens if the vertex-face incidence structure forms a partial geometric design. We also explore properties of vertex-face walks on the covers of a graph. Finally, we study a non-classical behavior of vertex-face walks.

References in corpus (5)

• Universal computation by quantum walk
• Equivalence of Szegedy's and Coined Quantum Walks
• Quantum Search on the Spatial Grid
• An Infinite Family of Circulant Graphs with Perfect State Transfer in Discrete Quantum Walks
• Real State Transfer

Cited by in corpus (1)

• Discrete-Time Quantum Walks and Graph Structures