We propose an intermediate walk continuously connecting an open quantum random walk and a quantum walk with parameters $M\in \mathbb{N}$ controlling a decoherence effect; if $M=1$, the walk coincides with an open quantum random walk, while $M=\infty$, the walk coincides with a quantum walk. We define a measure which recovers usual probability measures on $\mathbb{Z}$ for $M=\infty$ and $M=1$ and we observe intermediate behavior through numerical simulations for varied positive values $M$. In the case for $M=2$, we analytically show that a typical behavior of quantum walks appears even in a small gap of the parameter from the open quantum random walk. More precisely, we observe both the ballistically moving towards left and right sides and localization of this walker simultaneously. The analysis is based on Kato's perturbation theory for linear operator. We futher analyze this limit theorem in more detail and show that the above three modes are described by Gaussian distributions.
We consider a discrete time quantum walker in one dimension, where at each step, the step length $\ell$ is chosen from a distribution $P(\ell) \propto \ell^{-δ-1}$ with $\ell \leq \ell_{max}$. We evaluate the probability $f(x,t)$ that the walker is at position $x$ at time $t$ and its first two moments. As expected, the disorder effectively localizes the walk even for large values of $δ$. Asymptotically, $\langle x^2 \rangle \propto t^{3/2}$ and $\langle x \rangle \propto t^{1/2}$ independent of $δ$ and $\ell$, both finite. The scaled distribution $f(x,t)t^{1/2}$ plotted versus $x/t^{1/2}$ show…
We show that particle transport in a uniform, quantum multi-baker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semi-classical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time from diffusive to ballistic motion on the order of the inverse of Planck's constant. We can argue, that for a large class of 1D quantum random walks, sim…
The effect of unitary noise on the discrete one-dimensional quantum walk is studied using computer simulations. For the noiseless quantum walk, starting at the origin (n=0) at time t=0, the position distribution Pt(n) at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, sigma(t) scales as sigma(t) ~ t, unlike the classical random walk for which sigma(t) ~ sqrt{t}. It is shown that when the quantum walk is exposed to unitary noise, it exhibits a crossover from quantum behavior for short times to classical-like be…
In this paper, we consider the quantum walk on $\mathbb{Z}$ with attachment of one-length path periodically. This small modification to $\mathbb{Z}$ provides localization of the quantum walk. The eigenspace causing this localization is generated by finite length round trip paths. We find that the localization is due to the eigenvalues of an underlying random walk. Moreover we find that the transience of the underlying random walk provides a slow down of the pseudo velocity of the induced quantum walk and a different limit distribution from the Konno distribution.
There has recently been considerable interest in quantum walks in connection with quantum computing. The walk can be considered as a quantum version of the so-called correlated random walk. We clarify a strong structural similarity between both walks and study limit theorems and absorption problems for correlated random walks by our PQRS method, which was used in our analysis of quantum walks.
We derive an expression for the mean square displacement of a particle whose motion is governed by a uniform, periodic, quantum multi-baker map. The expression is a function of both time, $t$, and Planck's constant, $\hbar$, and allows a study of both the long time, $t\to\infty$, and semi-classical, $\hbar\to 0$, limits taken in either order. We evaluate the expression using random matrix theory as well as numerically, and observe good agreement between both sets of results. The long time limit shows that particle transport is generically ballistic, for any fixed value of Planck's constant. Ho…
We consider the dynamical properties of dissipative continuous-time quantum walks on directed graphs. Using a large-deviation approach we construct a thermodynamic formalism allowing us to define a dynamical order parameter, and to identify transitions between dynamical regimes. For a particular class of dissipative quantum walks we propose a quantum generalization of the the classical PageRank vector, used to rank the importance of nodes in a directed graph. We also provide an example where one can characterize the dynamical transition from an effective classical random walk to a dissipative …
Quantum walks have been shown to have impressive transport properties compared to classical random walks. However, imperfections in the quantum walk algorithm can destroy any quantum mechanical speed-up due to Anderson localization. We numerically study the effect of static disorder on a quantum walk on the glued trees graph. For small disorder, we find that the dominant effect is a type of quantum decay, and not quantum localization. For intermediate disorder, there is a crossover to diffusive transport, while a localization transition is observed at large disorder, in agreement with Anderson…
The dynamics of the survival probability of quantum walkers on a one-dimensional lattice with random distribution of absorbing immobile traps are investigated. The survival probability of quantum walkers is compared with that of classical walkers. It is shown that the time dependence of survival probability of quantum walkers has a piecewise stretched exponential character depending on the density of traps in numerical and analytical observations. The crossover between the quantum analogs of the Rosenstock and Donsker-Varadhan behaviors is identified.
A discrete time quantum walk is considered in which the step lengths are chosen to be either $1$ or $2$ with the additional feature that the walker is persistent with a probability $p$. This implies that with probability $p$, the walker repeats the step length taken in the previous step and is otherwise antipersistent. We estimate the probability $P(x,t)$ that the walker is at $x$ at time $t$ and the first two moments. Asymptotically, $\langle x^2 \rangle = t^ν$ for all $p$. For the extreme limits $p=0$ and $1$, the walk is known to show ballistic behaviour, i.e., $ν= 2$. As $p$ is varied from…
Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength W, where a quantum to classical crossover at time $t_{c}\propto W^{-2}$ transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution and examine in detail a …
A new model that maps a quantum random walk described by a Hadamard operator to a particular case of a random walk is presented. The model is represented by a Markov chain with a stochastic matrix, i.e., all the transition rates are positive, although the Hadamard operator contains negative entries. Using a proper transformation that is applied to the random walk distribution after n steps, the probability distributions in space of the two quantum states |1>, |0> are revealed. These show that a quantum walk can be entirely mapped to a particular case of a higher dimension of a random walk mode…
This work deals with both instantaneous uniform mixing property and temporal standard deviation for continuous-time quantum random walks on circles in order to study their fluctuations comparing with discrete-time quantum random walks, and continuous- and discrete-time classical random walks.
An atom-cavity system consists of an atom or group of atoms inside a cavity. When an atom in cavity is stimulated by a laser pump, it is affected by the atom-field interaction shows the quasi-random walk behavior. This can change the entanglement and trace distance of system. In this work, the change entanglement and trace distance of system in two different cases namely the random walk and non-random walk is considered. The descriptive system is a two-level atom in the electrodynamics cavity based on the Jaynes-Cummings model, which is stimulated by two longitudinal and transverse laser pumps…
We study the temporal growth of the von Neumann entropy for dissipative quantum walks on networks. By using a phenomenological quantum master equation, the quantum stochastic walk (QSW), we are able to parametrically scan the crossover from purely coherent quantum walks to purely diffusive random walks. In the latter limit the entropy shows a logarithmic growth, which is proportional to the information dimension of the random walk on the network. Here we present results for the von Neumann entropy based on the reduced density operator of the QSW. It shows a similar logarithmic growth for a wid…
We use simple deterministic dynamical systems as coins in studying quantum walks. These dynamical systems can be chosen to display, in the classical limit, a range of behaviors from the integrable to chaotic, or deterministically random. As an example of an integrable coin we study the Fourier walk that generalizes the Hadamard walk and show that the walker slows down with coin dimensionality, which controls the effective Planck constant. Introducing multi-Harper maps as deterministic models of random walks we study the effect of coin chaos on the quantum walk. We also demonstrate that breakin…
