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arXiv:2607.14797

Robustness of periodicity in Grover walks under a magnetic vector potential

Hiroto Sekido, Etsuo Segawa

quant-phmath-phmath.COSubmitted Jul 16, 2026• Announced Jul 17, 2026• 0 citations• Revised Sep 29, 2026 (v2)

v1 · Jul 16, 2026 v2 · Sep 29, 2026

Abstract

We study the effect of magnetic vector potentials on periodic Grover walks on finite graphs. The magnetic vector potential is introduced through the framework of quantum graphs, which induces the Grover walk as a special case. We regard the magnetic vector potential as a perturbation of a periodic Grover walk and investigate the robustness of its periodicity. Our analysis reveals that the response to such perturbations depends on the spectral structure of the underlying graph. In particular, when the graph possesses at least one non-simple eigenvalue, we derive a Hermitian matrix that characterizes the robustness of its periodicity. As a consequence, for initial states orthogonal to the eigenspaces of the unperturbed Grover walk corresponding to the eigenvalues $\pm1$, we show that the perturbed dynamics is asymptotically described by a continuous-time quantum walk generated by this Hermitian matrix.

Comments: 41 pages, 3 figures

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We study the effect of magnetic vector potentials on periodic Grover walks on finite graphs. The magnetic vector potential is introduced through the framework of quantum graphs, which induces the Grover walk as a special case. We regard the magnetic vector potential as a perturbation of a periodic Grover walk and investigate the robustness of its periodicity. Our analysis reveals that the response to such perturbations depends on the spectral structure of the underlying graph. In particular, when the graph possesses at least one non-simple eigenvalue, we derive a Hermitian matrix that characterizes the robustness of its periodicity. As a consequence, for initial states orthogonal to the eigenspaces of the unperturbed Grover walk corresponding to the eigenvalues

\pm1

, we show that the perturbed dynamics is asymptotically described by a continuous-time quantum walk generated by this Hermitian matrix.

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.

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, 4

, and

5

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.

This paper introduces a group-theoretic framework to analyze the algebraic structure of the Grover walk on a complete graph with self-loops. We construct a group generated by the Grover matrix and a diagonal matrix whose entries are powers of a complex root of unity. We then characterize the resulting quotient group, which is defined using a subgroup formed by commutators involving these matrices. We show that this quotient group is isomorphic to a finite cyclic group whose structure depends on the parity of the number of vertices. This group-theoretic characterization reveals underlying symmetries in the time evolution of the Grover walk and provides an algebraic framework for understanding its periodic behavior.

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.

We consider the Grover walk on a finite graph composed of two arbitrary simple graphs connected by one edge, referred to as a bridge. The parameter

ε>0

assigned at the bridge represents the strength of connectivity: if

ε=0

, then the graph is completely separated. We show that for sufficiently small values of

ε

, a phenomenon called pulsation occurs. The pulsation is characterized by the periodic transfer of the quantum walker between the two graphs. An asymptotic expression with respect to small

5

for the probability of finding the walker on either of the two graphs is derived. This expression reveals that the pulsation depends solely on the number of edges in each graph, regardless of their structure. In addition, we obtain that the quantum walker is transferred periodically between the two graphs, with a period of order

O(ε^{-1/2})

. Furthermore, when the number of edges of two graphs is equal, the quantum walker is almost completely transferred.