scholarly journals Eigenvalues of Two-State Quantum Walks Induced by the Hadamard Walk

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 127
Author(s):  
Shimpei Endo ◽  
Takako Endo ◽  
Takashi Komatsu ◽  
Norio Konno
Keyword(s):  
Numerical Simulation ◽  
Generating Function ◽  
Discrete Time ◽  
Function Method ◽  
Characteristic Parameter ◽  
Quantum Walks ◽  
One Dimension ◽  
Time Quantum ◽  
First Time ◽  
Inhomogeneous Quantum

Existence of the eigenvalues of the discrete-time quantum walks is deeply related to localization of the walks. We revealed, for the first time, the distributions of the eigenvalues given by the splitted generating function method (the SGF method) of the space-inhomogeneous quantum walks in one dimension we had treated in our previous studies. Especially, we clarified the characteristic parameter dependence for the distributions of the eigenvalues with the aid of numerical simulation.

scholarly journals Limit measures of inhomogeneous discrete-time quantum walks in one dimension

2012 ◽  
Vol 12 (1) ◽  
pp. 33-53 ◽  
Author(s):  
Norio Konno ◽  
Tomasz Łuczak ◽  
Etsuo Segawa
Keyword(s):  
Discrete Time ◽  
Quantum Walks ◽  
One Dimension ◽  
Time Quantum

scholarly journals Orthogonal Polynomials Induced by Discrete-Time Quantum Walks in One Dimension

2009 ◽  
Vol 15 (3) ◽  
pp. 367-375 ◽  
Author(s):  
Masatoshi HAMADA ◽  
Norio KONNO ◽  
Wojciech MŁOTKOWSKI
Keyword(s):  
Orthogonal Polynomials ◽  
Discrete Time ◽  
Quantum Walks ◽  
One Dimension ◽  
Time Quantum

scholarly journals Discrete-time quantum walks generated by aperiodic fractal sequence of space coin operators

2018 ◽  
Vol 29 (10) ◽  
pp. 1850098 ◽  
Author(s):  
R. F. S. Andrade ◽  
A. M. C. Souza
Keyword(s):  
Wave Function ◽  
Probability Distribution ◽  
Discrete Time ◽  
Numerical Study ◽  
Entanglement Entropy ◽  
Quantum Effects ◽  
Quantum Walks ◽  
Cantor Set ◽  
One Dimensional ◽  
Time Quantum

Properties of one-dimensional discrete-time quantum walks (DTQWs) are sensitive to the presence of inhomogeneities in the substrate, which can be generated by defining position-dependent coin operators. Deterministic aperiodic sequences of two or more symbols provide ideal environments where these properties can be explored in a controlled way. Based on an exhaustive numerical study, this work discusses a two-coin model resulting from the construction rules that lead to the usual fractal Cantor set. Although the fraction of the less frequent coin [Formula: see text] as the size of the chain is increased, it leaves peculiar properties in the walker dynamics. They are characterized by the wave function, from which results for the probability distribution and its variance, as well as the entanglement entropy, were obtained. A number of results for different choices of the two coins are presented. The entanglement entropy has shown to be very sensitive to uncovering subtle quantum effects present in the model.

scholarly journals Dynamics and energy spectra of aperiodic discrete-time quantum walks

2017 ◽  
Vol 96 (1) ◽  
Author(s):  
N. Lo Gullo ◽  
C. V. Ambarish ◽  
Th. Busch ◽  
L. Dell'Anna ◽  
C. M. Chandrashekar
Keyword(s):  
Discrete Time ◽  
Quantum Walks ◽  
Energy Spectra ◽  
Time Quantum

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

2015 ◽  
Vol 15 (11&12) ◽  
pp. 1060-1075
Author(s):  
Norio Konno ◽  
Masato Takei
Keyword(s):  
Eigenvalue Problem ◽  
Discrete Time ◽  
Quantum Walks ◽  
Stationary Measure ◽  
Unitary Matrix ◽  
Uniform Measure ◽  
Simple Argument ◽  
Stationary Measures ◽  
Time Quantum ◽  
The One

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.

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