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arXiv:2501.18600 · International Journal of Quantum Information

Periodicity and absolute zeta functions of multi-state grover walks on cycles

Jirô Akahori, Norio Konno, Iwao Sato, Yuma Tamura

Published Mar 1, 2025• 1 citation
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Abstract

Quantum walks, the quantum counterpart of classical random walks, are extensively studied for their applications in mathematics, quantum physics, and quantum information science. This study explores the periods and absolute zeta functions of Grover walks on cycle graphs. Specifically, we investigate Grover walks with an odd number of states and determine their periods for cycles with any number of vertices greater than or equal to two. In addition, we compute the absolute zeta functions of M-type Grover walks with finite periods. These results advance the understanding of the properties of Grover walks and their connection to absolute zeta functions.

DOI
10.1142/s0219749925500078

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Quantum walks, the quantum counterpart of classical random walks, are extensively studied for their applications in mathematics, quantum physics, and quantum information science. This study explores the periods and absolute zeta functions of Grover walks on cycle graphs. Specifically, we investigate Grover walks with an odd number of states and determine their periods for cycles with any number of vertices greater than or equal to two. In addition, we compute the absolute zeta functions of M-type Grover walks with finite periods. These results advance the understanding of the properties of Grover walks and their connection to absolute zeta functions.
The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over F1\mathbf{F}_1F1​. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and 333-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.
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.
The Grover algorithm is one of the most famous quantum algorithms. On the other hand, the absolute zeta function can be regarded as a zeta function over [Formula: see text] defined by a function satisfying the absolute automorphy. In this study, we show the property of the Grover algorithm and present a relation between the Grover algorithm and the absolute zeta function. We focus on the period of the Grover algorithm because if the period is finite, then we are able to get an absolute zeta function explicitly by Kurokawa’s theorem. In addition, whenever the period is finite or not, an expansion of the absolute zeta function can be obtained by a direct computation.
Abstract We consider the zeta function associated with the time evolution matrix of a bipartite walk on a bipartite graph and derive a formula for its absolute zeta function. Furthermore, we give formulas for the absolute zeta functions of the zeta functions of the time evolution matrices of bipartite walks on the complete bipartite graph and an even cycle graph. As an application, we obtain formulas for the absolute zeta functions of the zeta functions of the Grover matrices of the complete bipartite graph and an even cycle graph.
Dukes (2014) and Konno, Shimizu, and Takei (2017) studied the periodicity for 2-state quantum walks whose coin operator is the Hadamard matrix on cycle graph C_N with N vertices. The present paper treats the periodicity for 3-state quantum walks on C_N. Our results follow from a new method based on the cyclotomic field. This method gives a necessary condition for the coin operator for quantum walks to be periodic. Moreover, we reveal the period T_N of typical two kinds of quantum walks, the Grover and Fourier walks. We prove that both walks do not have any finite period except for N=3, in which case T_3=6 (Grover), =12 (Fourier).
In this paper, following the recent paper on Walk/Zeta Correspondence by the first author and his coworkers, we compute the zeta function for the three- and four-state quantum walk and correlated random walk, and the multi-state random walk on the one-dimensional torus by using the Fourier analysis. We deal with also the four-state quantum walk and correlated random walk on the two-dimensional torus. In addition, we introduce a new class of models determined by the generalized Grover matrix bridging the gap between the Grover matrix and the positive-support of the Grover matrix. Finally, we give a generalized version of the Konno-Sato theorem for the new class. As a corollary, we calculate the zeta function for the generalized Grover matrix on the d-dimensional torus.