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arXiv:1306.1807 · Physical Review A

Unidirectional quantum walks: Evolution and exit times

Miquel Montero

Published Jul 31, 2013• 18 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.88.012333

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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.
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.
We examine a discrete-time quantum walk with two-step memory for a particle on a onedimensional infinite space. The walk is defined with a four-state memory space analogous to the two-state coin space commonly used in discrete time quantum walks, and a method is presented for calculating the time evolution by using the Fourier transform. An integral expression for the probability is calculated, and this is used to produce numerical solutions for the probability distribution as a function of the time step and position. The results show two peaks in the probability distribution. One peak propagates ballistically with time, which is a common feature of quantum walks. The other peak is stationary with time and located at the initial site of the particle. This feature is not common in quantum walks and suggests that tracing the immediate history of the particle using two-step memory may represent the beginning of a transition to a classical system.
Abstract In the existing literature, a sojourn time of a discrete-time quantum walk is not a random variable. To solve this problem, we redefine the sojourn time of a quantum walk where its coin evolution operator can be general. We first discuss a class of quantum walks governed by flip operators. We cumulatively calculate how much time a walker spends in the set of non-negative integers up to a fixed evolution time. Whether a walker makes a left or right evolution, we add up the staying times as long as it stays within the target set. We define a sojourn time as the total amount of the staying times. Compared with existing definitions, we show that this definition can satisfy the probability normalization. From this, we define a random variable about the sojourn time and discuss its probability distribution. We build a mathematical model to characterize a sojourn time that is embedded into a quantum walk. These results are also valid for a class of quantum walks governed by general coin operators. We also give a method for calculating the sojourn time and analyze the shape features of its probability distribution.
The evolution of a walker in standard "Discrete-time Quantum Walk (DTQW)" is determined by coin and shift unitary operators. The conditional shift operator shifts the position of the walker to right or left by unit step size while the direction of motion is specified by the coin operator. This scenario can be generalized by choosing the step size randomly at each step in some specific interval. For example, the value of the roll of a dice can be used to specify the step size after throwing the coin. Let us call such a quantum walk "Discrete-time Random Step Quantum Walk (DTRSQW)". A completely random probability distribution is obtained whenever the walker follows the DTRSQW. We have also analyzed two more types of quantum walks, the "Discrete-time Un-biased Quantum Walk (DTUBQW)" and the "Discrete-time Biased Quantum Walk (DTBQW)". In the first type, the step size is kept different than unit size but the same for left and right shifts, whereas in the second type left and right shifts can also be different. The probability distribution in DTUBQW is found to follow a certain rule. The standard deviation (σσσ) of DTRSQW is higher than DTQW and hence DTRSQW spreads faster. The σσσ of DTUBQW shows sawtooth behavior with faster spread than DTQW for some specific values of rotation angles and steps.
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.
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.