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

arXiv:1111.6630 · Cambridge University Press eBooks

The Quantum Walk of F. Riesz

F. Alberto Grünbaum, L. Velázquez

Published Nov 15, 2012• 7 citations
arXivHTMLPDF

Abstract

We exhibit a way to associate a quantum walk (QW) on the non-negative integers to any probability measure on the unit circle. This forces us to consider one step transitions that are not traditionally allowed. We illustrate this in the case of a very interesting measure, originally proposed by F. Riesz for a different purpose.
DOI
10.1017/cbo9781139095402.006

Related papers

Return probability and self-similarity of the Riesz walk

Ryota Hanaoka, Norio Konno

The quantum walk is a counterpart of the random walk. The 2-state quantum walk in one dimension can be determined by a measure on the unit circle in the complex plane. As for the singular continuous measure, results on the corresponding quantum walk are limited. In this situation, we focus on a quantum walk, called the Riesz walk, given by the Riesz measure which is one of the famous singular continuous measures. The present paper is devoted to the return probability of the Riesz walk. Furthermore, we present some conjectures on the self-similarity of the walk.

arXiv:2010.04518•Quantum Information and Computation•Mar 22, 2021•1 citation

A discrete-time quantum walk taking continuous steps

Alex D. Gottlieb

This note introduces a quantum random walk in Euclidean space and proves the weak convergence of its rescaled n-step densities.

•Oct 4, 2003•0 citations

Matrix valued Szego polynomials and quantum random walks

Maŕıa José Cantero, F. Alberto Grünbaum, Moral, L. +1 more

We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non- negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being selfcontained, but we re- frain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 18]. See also the recent notes [19].

arXiv:0901.2244•Jan 15, 2009•1 citation

Quantization and asymptotic behaviour of quantum random walk on integers

Demosthenes Ellinas, Ioannis Smyrnakis

arXiv:quant-ph/0510098•Physica A Statistical Mechanics and its Applications•Feb 1, 2006•7 citations

On probability polynomials of 1D quantum walk

Mark W. Coffey, Matthew S. Heller

Quantum Information Processing•Aug 26, 2010•0 citations

Matrix‐valued Szegő polynomials and quantum random walks

Maŕıa José Cantero, L. Moral, F. Alberto Grünbaum +1 more

Abstract We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth‐and‐death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case, the study of QRWs on the set of nonnegative integers can be handled using scalar‐valued (Laurent) polynomials and a scalar‐valued measure on the circle. In the case of classical or quantum random walks on the integers, one needs to allow for matrix‐valued versions of these notions. We show how our tools yield results in the well‐known case of the Hadamard walk, but we go beyond this translation‐invariant model to analyze examples that are hard to analyze using other methods. More precisely, we consider QRWs on the set of nonnegative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being self‐contained, but we refrain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 9]. See also the recent notes [20]. © 2009 Wiley Periodicals, Inc.

Communications on Pure and Applied Mathematics•Dec 29, 2009•105 citations
The quantum walk is a counterpart of the random walk. The 2-state quantum walk in one dimension can be determined by a measure on the unit circle in the complex plane. As for the singular continuous measure, results on the corresponding quantum walk are limited. In this situation, we focus on a quantum walk, called the Riesz walk, given by the Riesz measure which is one of the famous singular continuous measures. The present paper is devoted to the return probability of the Riesz walk. Furthermore, we present some conjectures on the self-similarity of the walk.
We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non- negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being selfcontained, but we re- frain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 18]. See also the recent notes [19].
Abstract We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth‐and‐death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case, the study of QRWs on the set of nonnegative integers can be handled using scalar‐valued (Laurent) polynomials and a scalar‐valued measure on the circle. In the case of classical or quantum random walks on the integers, one needs to allow for matrix‐valued versions of these notions. We show how our tools yield results in the well‐known case of the Hadamard walk, but we go beyond this translation‐invariant model to analyze examples that are hard to analyze using other methods. More precisely, we consider QRWs on the set of nonnegative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being self‐contained, but we refrain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 9]. See also the recent notes [20]. © 2009 Wiley Periodicals, Inc.