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Journal of the Mathematical Society of Japan

Intertwining operators between one-dimensional space-homogeneous quantum walks

Hiroki Sako

Published Mar 9, 2026• 0 citations
Publisher

Abstract

The subject of this paper is a kind of dynamical systems called quantum walks. We study one-dimensional homogeneous analytic quantum walks $U$. We explain how to identify the space of all the uniform intertwining operators between these walks. We can also determine whether $U$ can be realized by a (not necessarily homogeneous) continuous-time uniform quantum walk on $\mathbb{Z}$. Several examples of quantum walks, which can not be realized by continuous-time uniform quantum walks, are presented. The 4-state Grover walk is one of them. Before stating the main theorems, we clarify the definition of one-dimensional quantum walks. For the first half of this paper, we study basic properties of one-dimensional quantum walks, which are not necessarily homogeneous. An equivalence relation between quantum walks called similarity is also introduced. This allows us to manipulate quantum walks in a flexible manner.

DOI
10.2969/jmsj/95459545

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The subject of this paper is a kind of dynamical systems called quantum walks. We study one-dimensional homogeneous analytic quantum walks UUU. We explain how to identify the space of all the uniform intertwining operators between these walks. We can also determine whether UUU can be realized by a (not necessarily homogeneous) continuous-time uniform quantum walk on Z\mathbb{Z}Z. Several examples of quantum walks, which can not be realized by continuous-time uniform quantum walks, are presented. The 4-state Grover walk is one of them. Before stating the main theorems, we clarify the definition of one-dimensional quantum walks. For the first half of this paper, we study basic properties of one-dimensional quantum walks, which are not necessarily homogeneous. An equivalence relation between quantum walks called similarity is also introduced. This allows us to manipulate quantum walks in a flexible manner.
The subject of this paper is a kind of dynamical systems called quantum walks. We study one-dimensional homogeneous analytic quantum walks U. We explain how to identify the space of all the uniform intertwining operators between these walks. We can also determine whether U can be realized by a (not necessarily homogeneous) continuous-time uniform quantum walk on Z. Several examples of quantum walks, which can not be realized by continuous-time uniform quantum walks, are presented. The 4-state Grover walk is one of them. Before stating the main theorems, we clarify the definition of one-dimensional quantum walks. For the first half of this paper, we study basic properties of one-dimensional quantum walks, which are not necessarily homogeneous. An equivalence relation between quantum walks called similarity is also introduced. This allows us to manipulate quantum walks in a flexible manner.
We propose categories of 111-dimensional and multi-dimensional quantum walks. In the categories, an object is a quantum walk, and a morphism is an intertwining operator between two quantum walks. The new framework enables us to discuss quantum walks in a unified way. The purposes of this paper are the following: (1) We reinterpret known results in our new framework. (2) We show several new theorems. For example, it is proved that every space-homogeneous time-periodic analytic quantum walk on Zd\mathbb{Z}^dZd has a limit distribution of velocity for every initial unit vector. Analyticity is a very weak condition. (3) We ask whether there exists a continuous-time quantum walk (V(t))t∈R(V^{(t)})_{t \in \mathbb{R}}(V(t))t∈R​ which realizes a given discrete-time quantum walk UUU. Existence of (V(t))t∈R(V^{(t)})_{t \in \mathbb{R}}(V(t))t∈R​ is equivalent to that of a 111-parameter group of automorphisms (V(t))t∈R(V^{(t)})_{t \in \mathbb{R}}(V(t))t∈R​ from the object UUU to UUU.
Space-homogeneous quantum walks on Z\mathbb{Z}Z from the viewpoint of complex analysis
The subject of this paper is quantum walks, which are expected to simulate several kinds of quantum dynamical systems. In this paper, we define analyticity for quantum walks on Z\mathbb{Z}Z. Almost all the quantum walks on Z\mathbb{Z}Z which have been already studied are analytic. In the framework of analytic quantum walks, we can enlarge the theory of quantum walks. We obtain not only several generalizations of known results, but also new types of theorems. It is proved that every analytic space-homogeneous quantum walk on Z\mathbb{Z}Z is essentially a composite of shift operators and continuous-time analytic space-homogeneous quantum walks. We also prove existence of the weak limit distribution for analytic space-homogeneous quantum walks on Z\mathbb{Z}Z.
The subject of this paper is quantum walks, which are expected to simulate several kinds of quantum dynamical systems. In this paper, we define analyticity for quantum walks on Z. Almost all the quantum walks on Z\mathbb{Z}Z which have been already studied are analytic. In the framework of analytic quantum walks, we can enlarge the theory of quantum walks. We obtain not only several generalizations of known results, but also new types of theorems. It is proved that every analytic space-homogeneous quantum walk on Z is essentially a composite of shift operators and continuous-time analytic space-homogeneous quantum walks. We also prove existence of the weak limit distribution for analytic space-homogeneous quantum walks on Z.
We undertake a detailed analysis of ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result of our paper holds in dimension one, and gives a complete equivalence between the absolutely continuous spectrum of the unitary operator encoding the walk, and the equidistribution of its dynamics in position space, which appears for the first time in the context of large-volume quantum ergodicity. In higher dimensions, we give a criterion for full and partial ergodicity in terms of a finer property of the spectrum which we dub ``No Repeating Graphs'', and we distinguish how strongly the equidistribution is taking place (weak convergence vs total variation). Many examples are included to illustrate the criterion and to distinguish between the types of ergodicity.
Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development of numerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. We revisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coin operator, constant in both space and time, and a localized initial state using numerical modeling. We investigate the impact of each parameter of the general coin operator on the probability distribution of the quantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution of the walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The study on general coin operators also includes the popular coins – Hadamard, Grover, and Fourier.