Discrete Time Quantum Walks Continuous Limit in 1 + 1 and 1 + 2 Dimension

2013 ◽  
Vol 10 (7) ◽  
pp. 1621-1625 ◽  
Author(s):  
Fabrice Debbasch ◽  
Giuseppe Di Molfetta
Keyword(s):  
Discrete Time ◽  
Quantum Walks ◽  
Continuous Limit ◽  
Time Quantum

scholarly journals Discrete Geometry from Quantum Walks

2019 ◽  
Vol 4 (2) ◽  
pp. 40 ◽  
Author(s):  
Fabrice Debbasch
Keyword(s):  
Discrete Geometry ◽  
Discrete Time ◽  
Quantum Walks ◽  
Space Time ◽  
Continuous Limit ◽  
Spin Connection ◽  
Riemann Curvature ◽  
Curved Space ◽  
Covariant Derivatives ◽  
Time Quantum

A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in 2D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant derivatives along the lattice coordinates and discrete connections. The concepts of metrics and 2-beins are also extended to the discrete realm. Two slightly different Riemann curvatures are then defined on the space-time lattice as the curvatures of the discrete spin connection. These two curvatures are closely related and one of them tends at the continuous limit towards the usual, continuous Riemann curvature. A simple example is also worked out in full.

scholarly journals General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

2020 ◽  
Vol 19 (10) ◽  
Author(s):  
Michael Manighalam ◽  
Mark Kon
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Time Limit ◽  
Continuous Limit ◽  
Continuous Space ◽  
Time Quantum ◽  
Space Limit

Abstract Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

scholarly journals Discrete-time quantum walks: Continuous limit and symmetries

2012 ◽  
Vol 53 (12) ◽  
pp. 123302 ◽  
Author(s):  
G. di Molfetta ◽  
F. Debbasch
Keyword(s):  
Discrete Time ◽  
Quantum Walks ◽  
Continuous Limit ◽  
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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