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A quantum walk on a line and a Weyl equation in a space

Takuya Machida

Published Apr 14, 2017• 0 citations
OpenAlex

Abstract

Discrete-time quantum walks are considered a discretization of Dirac equations and the study for them has been getting attention since around 2000. In this paper, we focus on a quantum walk on a line related to a Weyl equation in a space. The quantum walker with two inner states, also interpreted as a quantum particle with the down-spin and the up-spin state in physics, spreads on the line and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

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Discrete-time quantum walks are considered a discretization of Dirac equations and the study for them has been getting attention since around 2000. In this paper, we focus on a quantum walk on a line related to a Weyl equation in a space. The quantum walker with two inner states, also interpreted as a quantum particle with the down-spin and the up-spin state in physics, spreads on the line and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.
Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.
In this paper we focus our attention on a particle that follows a unidirectional quantum walk, an alternative version of the currently widespread discrete-time quantum walk on a line. Here the walker at each time step can either remain in place or move in a fixed direction, e.g., rightward or upward. While both formulations are essentially equivalent, the present approach leads us to consider discrete Fourier transforms, which eventually results in obtaining explicit expressions for the wave functions in terms of finite sums and allows the use of efficient algorithms based on the fast Fourier transform. The wave functions here obtained govern the probability of finding the particle at any given location but determine as well the exit-time probability of the walker from a fixed interval, which is also analyzed.
We consider a discrete-time 2-state quantum walk on the line. The state of the quantum walker evolves according to a rule which is determined by a coin-flip operator and a position-shift operator. In this paper we take a 3-periodic time evolution as the rule. For such a quantum walk, we get a limit distribution which expresses the asymptotic behavior of the walker after a long time. The limit distribution is different from that of a time-independent quantum walk or a 2-period time-dependent quantum walk. We give some analytical results and then consider a number of variants of our model and indicate the result of simulations for these ones.
The probability distributions of discrete-time quantum walks have been often investigated, and many interesting properties of them have been discovered. The probability that the walker can be find at a position is defined by diagonal elements of the density matrix. On the other hand, although off-diagonal parts of the density matrices have an important role to quantify quantumness, they have not received attention in quantum walks. We focus on the off-diagonal parts of the density matrices for discrete-time quantum walks on the line and derive limit theorems for them.