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arXiv:2105.04056 · Quantum Information and Computation

IPS/Zeta correspondence

Takashi Komatsu, Norio Konno, Iwao Sato

Published Feb 1, 2022• 3 citations
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Abstract

Our previous works presented zeta functions by the Konno-Sato theorem or the Fourier analysis for one-particle models including random walks, correlated random walks, quantum walks, and open quantum random walks. This paper introduces a new zeta function for multi-particle models with probabilistic or quantum interactions, called the interacting particle system (IPS). We compute the zeta function for some tensor-type IPSs.

DOI
10.26421/qic22.3-4-4

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Our previous works presented zeta functions by the Konno-Sato theorem or the Fourier analysis for one-particle models including random walks, correlated random walks, quantum walks, and open quantum random walks. This paper introduces a new zeta function for multi-particle models with probabilistic or quantum interactions, called the interacting particle system (IPS). We compute the zeta function for some tensor-type IPSs.
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.
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.
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.
Interacting particle systems studied in this paper are probabilistic cellular automata with nearest-neighbor interaction including the Domany-Kinzel model. A special case of the Domany-Kinzel model is directed percolation. We regard the interacting particle system as a Markov chain on a graph. Then we present a new quantization of the interacting particle system. After that, we introduce a zeta function of the quantized model and give its determinant expression. Moreover, we calculate the absolute zeta function of the quantized model for the Domany-Kinzel model.