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InTech eBooks

Quantum Walks

Takuya Machida

Published Aug 24, 2016• 1 citation
Publisher

Abstract

Quantum walks are quantum counterparts of random walks. While probability distributions of random walks diffusively spread out as the walkers are updating, quantum walks have ballistic behavior. Some of the ballistic behaviors have been revealed in long-time limit theorems and their probability distributions are all far away from the Gaussian distributions, which are known as limit distributions of random walks. In this chapter, we are going to be seeing limit distributions for a standard quantum walk on the line and two kinds of time-dependent quantum walk on the line.

DOI
10.5772/62481

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Quantum walks are quantum counterparts of random walks. While probability distributions of random walks diffusively spread out as the walkers are updating, quantum walks have ballistic behavior. Some of the ballistic behaviors have been revealed in long-time limit theorems and their probability distributions are all far away from the Gaussian distributions, which are known as limit distributions of random walks. In this chapter, we are going to be seeing limit distributions for a standard quantum walk on the line and two kinds of time-dependent quantum walk on the line.
It has been discovered that open quantum walks diffusively distribute in space, since they were introduced in 2012. Indeed, some limit distributions have been demonstrated and most of them are described by Gaussian distributions. We operate an open quantum walk on [Formula: see text] with parameterized operations in this paper, and study its 1st and 2nd moments so that we find its standard deviation. The standard deviation tells us whether the open quantum walker shows diffusive or ballistic behavior, which results in a phase transition of the walker.
When confined to a topological environment consisting of a cycle coupled with a half-line, quantum walks exhibit long-term statistical tendencies which differ dramatically from the tendencies of classical random walks in the same environment. In particular, as suggested by numerical simulations, the probability distribution of the walker's position resolves, in part, into a non-vanishing distribution on the cycle and, in part, into a ballistic distribution on the half-line. By contrast, for a classical random walk, the probability distribution of the walker's position tends always to vanish on the cycle and to migrate completely to the half-line as a purely diffusive process.
Quantum walks act in obviously different ways from their classical counterparts, but decoherence will lessen and close this gap between them. To understand this process, it is necessary to investigate the evolution of quantum walks under different decoherence situations. In this article, we study a non-Markovian decoherent quantum walk on a line. In a short time regime, the behavior of the walk deviates from both ideal quantum walks and classical random walks. The position variance as a measure of the quantum walk collapses and revives for a short time, and tends to have a linear relation with time. That is, the walker's behavior shows a diffusive spread over a long time limit, which is caused by non-Markovian dephasing affecting the quantum correlations between the quantum walker and his coin. We also study both quantum discord and measurement-induced disturbance as measures of the quantum correlations, and observe both collapse and revival in the short time regime, and the tendency to be zero in the long time limit. Therefore, quantum walks with non-Markovian decoherence tend to have diffusive spreading behavior over long time limits, while in the short time regime they oscillate between ballistic and diffusive spreading behavior, and the quantum correlation collapses and revives due to the memory effect.
Abstract We analyze the probability distributions of the quantum walks induced from Markov chains by Szegedy (2004). The first part of this paper is devoted to the quantum walks induced from finite state Markov chains. It is shown that the probability distribution on the states of the underlying Markov chain is always convergent in the Cesaro sense. In particular, we deduce that the limiting distribution is uniform if the transition matrix is symmetric. In the case of a non-symmetric Markov chain, we exemplify that the limiting distribution of the quantum walk is not necessarily identical with the stationary distribution of the underlying irreducible Markov chain. The Szegedy scheme can be extended to infinite state Markov chains (random walks). In the second part, we formulate the quantum walk induced from a lazy random walk on the line. We then obtain the weak limit of the quantum walk. It is noted that the current quantum walk appears to spread faster than its counterpart-quantum walk on the line driven by the Grover coin discussed in literature. The paper closes with an outlook on possible future directions.
Open Quantum Random Walks are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.
Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without losing coherence as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.