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Register a new userGrover/Zeta Correspondence based on the Konno-Sato theorem
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Recently the Ihara zeta function for the finite graph was extended to infinite one by Clair and Chinta et al. In this paper, we obtain the same expressions by a different approach from their analytical method. Our new approach is to take a suitable limit of a sequence of finite graphs via the Konno-Sato theorem. This theorem is related to explicit formulas of characteristic polynomials for the evolution matrix of the Grover walk. The walk is one of the most well-investigated quantum walks which are quantum counterpart of classical random walks. We call the relation between the Grover walk and the zeta function based on the Konno-Sato theorem Grover/Zeta Correspondence here.
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Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno-Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis.
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 presents a zeta function for multi-particle models with probabilistic or quantum interactions, called the interacting particle system (IPS). The zeta function for the tensor-type IPS is computed.
In our previous work, we investigated the relation between zeta functions and discrete-time models including random and quantum walks. In this paper, we introduce a zeta function for the continuous-time model (CTM) and consider CTMs including the corresponding random and quantum walks on the d-dimensional torus.
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.
We study the connection between mutually unbiased bases and mutually orthogonal extraordinary supersquares, a wider class of squares which does not contain only the Latin squares. We show that there are four types of complete sets of mutually orthogonal extraordinary supersquares for the dimension $d=8$. We introduce the concept of physical striation and show that this is equivalent to the extraordinary supersquare. The general algorithm for obtaining the mutually unbiased bases and the physical striations is constructed and it is shown that the complete set of mutually unbiased physical striations is equivalent to the complete set of mutually orthogonal extraordinary supersquares. We apply the algorithm to two examples: one for two-qubit systems ($d=4$) and one for three-qubit systems ($d=8$), by using the Type II complete sets of mutually orthogonal extraordinary supersquares of order 8.
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