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arXiv:1902.02479

Intertwining operators between one-dimensional homogeneous quantum walks

Hiroki Sako

Published Feb 7, 2019• 1 citation
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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 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
10.48550/arxiv.1902.02479

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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 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.
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 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.
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 Grover walk, which is related to Grover's search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is strikingly different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.