Quantum Walks on Embeddings

2017/11/23 by Hanmeng Zhan, Zhan, Hanmeng · 2 citations
Computer Science · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Physics (quant-ph) #Quantum-Dot Cellular Automata

paper · pdf · doi:10.48550/arxiv.1711.08831

openalex publication_date 2017/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 U², 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.

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• Pretty good state transfer in Grover walks on abelian Cayley graphs
• Perfect state transfer in Grover walks on association schemes and distance-regular graphs
• Discrete-Time Quantum Walks and Graph Structures

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