arXiv:2112.01000

A remark on uniform Strichartz estimate for quantum walks on 1D lattice

Aso, Takumi, Masaya Maeda

Published Dec 2, 2021• 0 citations

Abstract

In this short note, we study quantum walks (QWs) on one dimensional lattice $δ\mathbb{Z}$. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width $δ$.

DOI: 10.48550/arxiv.2112.01000

Related papers

The non-uniform stationary measure for discrete-time quantum walks in one dimension

KonnoNorio, TakeiMasato

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 × 2 unitary matrix U. In our previous paper [15], we proved th...

Quantum Information and ComputationSep 1, 20150 citations

Dispersive estimates for quantum walks on 1D lattice

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

arXiv:1808.05714Journal of the Mathematical Society of JapanAug 5, 202112 citations

Scattering and inverse scattering for nonlinear quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations and discrete nonlinear Schrödinger equations but it seems to be the first time to be applied to QWs.

arXiv:1801.03214Discrete and Continuous Dynamical SystemsJan 1, 201825 citations

DYNAMICAL LOCALIZATION OF RANDOM QUANTUM WALKS ON THE LATTICE

Alain Joye

This note describes recent results on the localization properties of Random Quantum Walks on the d-dimensional lattice in a regime analogous to the large disorder regime by means of the Fractional Moments Method adapted to the unitary framework.

arXiv:1209.4190WORLD SCIENTIFIC eBooksOct 1, 201311 citations

Limit Theorems for the Disordered Quantum Walk

Clement Boateng Ampadu

We study the disordered quantum walk in one dimension, and obtain the weak limit theorem.

arXiv:1108.6110Aug 31, 20110 citations

The non-uniform stationary measure for discrete-time quantum walks in one dimension

Norio Konno, Masato Takei

Quantum Information and ComputationSep 1, 201513 citations

arXiv:2112.01000

A remark on uniform Strichartz estimate for quantum walks on 1D lattice

Aso, Takumi, Masaya Maeda

Published Dec 2, 2021• 0 citations

arXiv HTML PDF

Abstract

In this short note, we study quantum walks (QWs) on one dimensional lattice

δ\mathbb{Z}

. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width

δ

In this short note, we study quantum walks (QWs) on one dimensional lattice $δ\mathbb{Z}$. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width $δ$.

DOI: 10.48550/arxiv.2112.01000

Related papers

The non-uniform stationary measure for discrete-time quantum walks in one dimension

KonnoNorio, TakeiMasato

Quantum Information and ComputationSep 1, 20150 citations

Dispersive estimates for quantum walks on 1D lattice

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$ . The dispersive estimate $\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_{c}$ is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$. The dispersive estimate $\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_{c}$ is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.

arXiv:1808.05714Journal of the Mathematical Society of JapanAug 5, 202112 citations

Scattering and inverse scattering for nonlinear quantum walks

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa +2 more

arXiv:1801.03214Discrete and Continuous Dynamical SystemsJan 1, 201825 citations

DYNAMICAL LOCALIZATION OF RANDOM QUANTUM WALKS ON THE LATTICE

Alain Joye

arXiv:1209.4190WORLD SCIENTIFIC eBooksOct 1, 201311 citations

Limit Theorems for the Disordered Quantum Walk

Clement Boateng Ampadu

We study the disordered quantum walk in one dimension, and obtain the weak limit theorem.

arXiv:1108.6110Aug 31, 20110 citations

The non-uniform stationary measure for discrete-time quantum walks in one dimension

Norio Konno, Masato Takei

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 $\times$ 2 unitary matrix $U$ . In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when $U$ is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 $\times$ 2 unitary matrix $U$. In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when $U$ is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.

Quantum Information and ComputationSep 1, 201513 citations

In this short note, we study quantum walks (QWs) on one dimensional lattice

δ\mathbb{Z}

. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width

δ

We consider quantum walks with position dependent coin on 1D lattice

\mathbb{Z}

. The dispersive estimate

\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}

is shown under

l^{1,1}

perturbation for the generic case and

l^{1,2}

perturbation for the exceptional case, where

U

is the evolution operator of a quantum walk and

P_{c}

is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2

\times

2 unitary matrix

U

. In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when

U

is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.