Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Mathematical Society of Japan

Space-homogeneous quantum walks on $\mathbb{Z}$ from the viewpoint of complex analysis

Hayato Saigo, Hiroki Sako

Published Aug 16, 2020• 4 citations
Publisher

Abstract

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 $\mathbb{Z}$. Almost all the quantum walks on $\mathbb{Z}$ 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 $\mathbb{Z}$ 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 $\mathbb{Z}$.

DOI
10.2969/jmsj/82648264

Related papers

Space-homogeneous quantum walks on Z from the viewpoint of complex analysis

Hayato Saigo, Hiroki Sako

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 $\mathbb{Z}$ 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.

arXiv:1802.01837•Feb 6, 2018•0 citations

Intertwining operators between one-dimensional space-homogeneous quantum walks

Hiroki Sako

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.

Journal of the Mathematical Society of Japan•Mar 9, 2026•0 citations

Categories of quantum walks

Hiroki Sako

We propose categories of $1$-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 $\mathbb{Z}^d$ 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 $(V^{(t)})_{t \in \mathbb{R}}$ which realizes a given discrete-time quantum walk $U$. Existence of $(V^{(t)})_{t \in \mathbb{R}}$ is equivalent to that of a $1$-parameter group of automorphisms $(V^{(t)})_{t \in \mathbb{R}}$ from the object $U$ to $U$.

arXiv:2003.12732•Mar 28, 2020•0 citations

Intertwining operators between one-dimensional homogeneous quantum walks

Hiroki Sako

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.

arXiv:1902.02479•Feb 7, 2019•1 citation

Convergence theorems on multi-dimensional homogeneous quantum walks

Hiroki Sako

Abstract We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved.

arXiv:1903.09342•Quantum Information Processing•Mar 1, 2021•5 citations

CONTINUOUS-TIME QUANTUM WALKS ON TREES IN QUANTUM PROBABILITY THEORY

Norio Konno

A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.

arXiv:quant-ph/0602213•Infinite Dimensional Analysis Quantum Probability and Related Topics•Jun 1, 2006•45 citations
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.
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.
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.
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.
Abstract We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved.