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arXiv:2211.02752 · Physica Scripta

Periodicity of bipartite walk on biregular graphs with conditional spectra

Qiuting Chen

Published Aug 21, 2024• 2 citations
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Abstract

Abstract In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover’s walks. A discrete quantum walk is given by the powers of a unitary matrix U indexed by arcs or edges of the underlying graph. The walk is periodic if U k = I for some positive integer k. Kubota has given a characterization of periodicity of Grover’s walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota’s results—if a biregular graph G has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over G in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover’s walk on regular graphs.

DOI
10.1088/1402-4896/ad71ff

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math.COCross-listedarXiv:2609.08315•Sep 9, 2026
Abstract In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover’s walks. A discrete quantum walk is given by the powers of a unitary matrix U indexed by arcs or edges of the underlying graph. The walk is periodic if U k = I for some positive integer k. Kubota has given a characterization of periodicity of Grover’s walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota’s results—if a biregular graph G has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over G in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover’s walk on regular graphs.
Recently, a research on quantum walks has been developed in various areas.In this paper we focus on the periodicity of the Grover walk which is one of the quantum walks on the discrete graphs.Then we find some special graphs to induce a periodic Grover walk: At some time k, the quantum state φ k returns its initial quantum state φ 0 .Our purpose is to characterize graphs which induce a k-periodic Grover walk for a fixed integer k.We do it for k = 2, 3, 4, 5 and gain a necessary condition for odd k.
Quantum walks determined by the coin operator on graphs have been intensively studied. The typical examples of coin operator are the Grover and Fourier matrices. The periodicity of the Grover walk is well investigated. However, the corresponding result on the Fourier walk is not known. In this paper, we present a necessary condition for the Fourier walk on regular graphs to have the finite period. As an application of our result, we show that the Fourier walks do not have any finite period for some classes of regular graphs such as complete graphs, cycle graphs with selfloops, and hypercubes.
Characterizations graphs of some classes to induce periodic Grover walks have been studied for recent years. In particular, for the strongly regular graphs, it has been known that there are only three kinds of such graphs. Here, we focus on the periodicity of the Grover walks on distance-regular graphs. The distance-regular graph can be regarded as a kind of generalization of the strongly regular graphs and the typical graph with an equitable partition. In this paper, we find some classes of such distance-regular graphs and obtain some useful necessary conditions to induce periodic Grover walks on the general distance-regular graphs. Also, we apply this necessary condition to give another proof for the strong regular graphs.
The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices uuu and vvv are adjacent if gcd⁡(u−v,n)=1\gcd(u - v, n) = 1gcd(u−v,n)=1. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs.
The Grover walk is one of the most well-studied quantum walks on graphs. In this paper, we investigate its periodicity to reveal the relationship between the quantum walk and the underlying graph, focusing particularly on the characterization of graphs exhibiting a periodic Grover walk. Graphs having a periodic Grover walk with periods of 2,3,42, 3, 42,3,4, and 555 have previously been characterized. It is expected that graphs exhibiting a periodic Grover walk with odd period correspond to cycles with odd length. We address that problem and are able to perfectly characterize the class of graphs exhibiting an odd-periodic Grover walk by using a combinatorial method.
Spectral characterization of the uniform theta graph Θ(t,2)Θ(t,2)Θ(t,2) and classification of 6-periodic Grover walks
We characterize the uniform theta graph Θ(t,2)Θ(t,2)Θ(t,2) by the spectrum of its normalized adjacency matrix, or equivalently, by the spectrum of its normalized Laplacian matrix. We also investigate the periodicity of Grover walks on nonregular graphs, which is closely related to the eigenvalues of the normalized adjacency matrix and those of the time evolution matrix of the Grover walk. We show that the Dutch windmill graph Dn(t)D_n^{(t)}Dn(t)​ is 2n2n2n-periodic and that the uniform theta graph Θ(t,n)Θ(t,n)Θ(t,n) is (2n+2)(2n+2)(2n+2)-periodic. Furthermore, we completely determine the connected 666-periodic graphs and prove that they are precisely D3(t)D_3^{(t)}D3(t)​ with t≥2t \geq 2t≥2 and Θ(t,2)Θ(t,2)Θ(t,2) with t≥1t \geq 1t≥1.