Perfect state transfer in Grover walks on association schemes and distance-regular graphs

2025/06/09 by Koushik Bhakta, Bikash Bhattacharjya, Bhakta, Koushik +1 · 1 citation
Computer Science · Mathematics · #05C50 #05E30 #81Q99 #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.2506.07439

openalex publication_date 2025/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper investigates perfect state transfer in Grover walks, a model of discrete-time quantum walks. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on graphs belonging to an association scheme. Our focus includes specific association schemes, namely the Hamming and Johnson schemes. We characterize all graphs on the classes of Hamming and Johnson schemes that exhibit perfect state transfer. Furthermore, we study perfect state transfer on distance-regular graphs. We provide complete characterizations for exhibiting perfect state transfer on distance-regular graphs of diameter 2 and diameter 3, as well as integral distance-regular graphs.

Citations

• State transfer in Grover walks on unitary and quadratic unitary Cayley graphs over finite commutative rings
• State transfer in discrete-time quantum walks via projected transition matrices
• Periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs
• Grover walks on unitary Cayley graphs and integral regular graphs
• Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks
• Regular graphs to induce even periodic Grover walks
• No perfect state transfer in trees with more than 3 vertices
• Hamiltonians of Bipartite Walks
• Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues
• Perfect state transfer in Grover walks between states associated to vertices of a graph
• Odd-periodic Grover walk
• Pretty good state transfer in discrete-time quantum walks
• Localization of two-dimensional quantum walks defined by generalized Grover coins
• Periodicity of Grover walks on distance-regular graphs
• Periodicity of Grover walks on generalized Bethe trees
• Quantum Walks on Embeddings
• Perfect quantum state transfer on the Johnson scheme
• An Infinite Family of Circulant Graphs with Perfect State Transfer in Discrete Quantum Walks
• Discrete-Time Quantum Walks and Graph Structures
• State Transfer on Graphs

Cited by

• Pretty good state transfer in Grover walks on abelian Cayley graphs

Related

• Perfect state transfer in Grover walks between states associated to vertices of a graph
• Quantum walks on simplexes and multiple perfect state transfer
• Strongly regular and strongly walk-regular graphs that admit perfect state transfer
• An Infinite Family of Circulant Graphs with Perfect State Transfer in Discrete Quantum Walks
• State transfer in discrete-time quantum walks via projected transition matrices