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Norio Konno

Publications and source records attributed to Norio Konno.

At least 19 recordsLinked to original sources
arXiv2025

Spectral analysis of hierarchical continuous-time quantum walks

In this paper, we introduce hierarchical random walks at first. In this model, we use two types of random walkers, {global and local} walkers. The global walker chooses a local walker at every step, then the chosen local walker moves a single step. After that we construct the corresponding continuous-time quantum walks and discuss its spectral structures. Then we define multi-dimensional continuous-time quantum walk by taking a marginal distribution respect to the global walker.

quant-ph↗
arXiv2025

Linear extrapolation for the graph of function of single variable based on walks

The quantum walk was introduced as a quantum counterpart of the random walk and has been intensively studied since around 2000. Its applications include topological insulators, radioactive waste reduction, and quantum search. The first author in 2019 defined a time-series model based on the measure of the ``discrete-time" and ``discrete-space" quantum walk in one dimension. Inspired by his model, this paper proposes a new model for the graph of a function of a single variable determined by the measure which comes from the weak limit measure of a ``continuous-time or discrete-time" and ``discrete-space" walk. The measure corresponds to a ``continuous-time" and ``continuous-space" walk in one dimension. Moreover, we also presents a method of a linear extrapolation for the graph by our model.

quant-ph↗
arXiv2025

Grover algorithm and absolute zeta functions

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 $\mathbb{F}_{1}$ 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.

quant-ph↗
arXiv2025

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

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.

quant-ph↗
arXiv2024

Absolute zeta functions for zeta functions of quantum walks

This paper presents a connection between the quantum walk and the absolute mathematics. The quantum walk is a quantum counterpart of the classical random walk. We especially deal with the Grover walk on a graph. The Grover walk is a typical model of quantum walks. The time evolution of the Grover walk is obtained by a unitary matrix that is called the Grover matrix. We define the zeta function determined by the Grover matrix. First we prove that the zeta function of the Grover walk is the absolute automorphic form that constructs the absolute zeta function. Next we calculate the absolute zeta function defined by Grover walks on some graphs. The absolute zeta functions of the Grover walks are expressed by the multiple gamma function. Other types of absolute zeta functions are obtained as an analogue of the multiple gamma function.

quant-ph↗
arXiv2024

Multi-dimensional continuous time quantum walks related to the birth and death chains

In this paper, we consider multi-dimensional birth and death chains and continuous time quantum walks (CTQW) related to them. For CTQW related to our forms of multi-dimensional birth and death chains, we obtain the time scaled independence between multiple dimensions about the transition probability of CTQW. By using this feature, we analyze CTQW on the path graph, which is related to 1-dimensional Ehrenfest model. We also have a random variable which is related to our models and converges to the standard Gaussian distribution.

quant-ph↗
arXiv2024

On the relation between quantum walks and absolute zeta functions

The quantum walk is a quantum counterpart of the classical random walk. On the other hand, the absolute zeta function can be considered as a zeta function over F_1. This paper presents a connection between the quantum walk and the absolute zeta function. First we deal with a zeta function determined by a time evolution matrix of the Grover walk on a graph. The Grover walk is a typical model of the quantum walk. Then we prove that the zeta function given by the quantum walk is an absolute automorphic form of weight depending on the number of edges of the graph. Furthermore we consider an absolute zeta function for the zeta function based on a quantum walk. As an example, we compute an absolute zeta function for the cycle graph and show that it is expressed as the multiple gamma function of order 2.

quant-ph↗
arXiv2024

Absolute zeta functions and periodicity of quantum walks on cycles

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 $\mathbb{F}_1$. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and $3$-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.

quant-ph↗
arXiv2024

A quantization of interacting particle systems

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.

quant-ph↗
arXiv2024

QW-Search/Zeta Correspondence

We consider the connection between this zeta function and quantum search via quantum walk. First, we give an explicit expression of the zeta function on the one-dimensional torus in the general case of the number and position of marked vertices. Moreover, we deal with the two special cases of the position of the marked vertices on the $d$-dimensional torus $(d \ge 2)$. Additionally, we treat the property of the zeta function by using the Mahler measure. Our results show the relationship between the zeta function and quantum search algorithms for the first time.

quant-ph↗
arXiv2024

Absolute zeta functions for zeta functions of quantum cellular automata

Our previous work dealt with the zeta function for the interacting particle system (IPS) including quantum cellular automaton (QCA) as a typical model in the study of ``IPS/Zeta Correspondence". On the other hand, the absolute zeta function is a zeta function over F_1 defined by a function satisfying an absolute automorphy. This paper proves that a new zeta function given by QCA is an absolute automorphic form of weight depending on the size of the configuration space. As an example, we calculate an absolute zeta function for a tensor-type QCA, and show that it is expressed as the multiple gamma function. In addition, we obtain its functional equation by the multiple sine function.

quant-ph↗
arXiv2023

Spectral analysis of three-state quantum walks with general coin matrices

Mathematical analysis of the spectral properties of the time evolution operator in quantum walks is essential for understanding key dynamical behaviors such as localization and long-term evolution. The inhomogeneous three-state case, in particular, poses substantial analytical challenges due to its higher internal degrees of freedom and the absence of translational invariance. We develop a general framework for the spectral analysis of three-state quantum walks on the one-dimensional lattice with arbitrary time evolution operators. Our approach is based on a transfer matrix formulation that reduces the infinite-dimensional eigenvalue problem to a tractable system of two-dimensional recursions, enabling exact characterization of eigenstates. This framework applies broadly to space-inhomogeneous models, including those with finite defects and two-phase structures. We rigorously derive necessary and sufficient conditions for the existence of point spectrum, along with a complete description of the corresponding eigenvalues and eigenstates, which are known to underlie quantum localization phenomena. Furthermore, we give a complete spectral decomposition -- discrete spectrum, flat-band eigenvalues (of infinite multiplicity), and absolutely continuous spectrum -- with explicit characterization of each component. Using this method, we perform exact numerical analyses of the Fourier walk with spatial inhomogeneity, revealing the emergence of localization despite its delocalized nature in the homogeneous case. Our results provide mathematical tools and physical insights into the structure of quantum walks, offering a systematic path for identifying and characterizing localized quantum states in complex quantum systems.

quant-ph↗
arXiv2023

Parrondo's game of quantum search based on quantum walk

The Parrondo game, devised by Parrondo, means that winning strategy is constructed a combination of losing strategy. This situation is called the Parrondo paradox. The Parrondo game based on quantum walk and the search algorithm via quantum walk have been widely studied, respectively. This paper newly presents a Parrondo game of quantum search based on quantum walk by combining both models. Moreover we confirm that Parrondo's paradox exists for our model on the one- and two-dimensional torus by numerical simulations. Afterwards we show the range in which the paradox occurs is symmetric about the origin on the $d$-dimensional torus $(d \geq 1)$ with even vertices and one marked vertex.

quant-ph↗
arXiv2023

Alternating Walk/Zeta Correspondence

We consider the alternating zeta function and the alternating $L$-function of a graph $G$, and express them by using the Ihara zeta function of $G$. Next, we define a generalized alternating zeta function of a graph, and express the generalized alternating zeta function of a vertex-transitive regular graph by spectra of the transition probability matrix of the symmetric simple random walk on it and its Laplacian. Furthermore, we present an integral expression for the limit of the generalized alternating zeta functions of a series of vertex-transitive regular graphs. As an example, we treat the generalized alternating zeta functions of a finite torus. Finally, we treat the relation between the Mahler measure and the alternating zeta function of a graph.

math.CO↗
arXiv2023

Ronkin/Zeta Correspondence

The Ronkin function was defined by Ronkin in the consideration of the zeros of almost periodic function. Recently, this function has been used in various research fields in mathematics, physics and so on. Especially in mathematics, it has a closed connections with tropical geometry, amoebas, Newton polytopes and dimer models. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on Zeta Correspondence. The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Ronkin function and our zeta function for random walks and quantum walks. Firstly we consider this relation in the case of one-dimensional random walks. Afterwards we deal with higher-dimensional random walks. For comparison with the case of the quantum walk, we also treat the case of one-dimensional quantum walks. Our results bridge between the Ronkin function and the zeta function via quantum walks for the first time.

math-ph↗
arXiv2022

Walk/Zeta Correspondence

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.

quant-ph↗
arXiv2022

The limit theorem with respect to the matrices on non-backtracking paths of a graph

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

math.CO↗