One-dimensional lazy quantum walks and occupancy rate

Physical Review A, 2020

Open quantum walks (OQWs) describe a quantum walker on an underlying graph whose dynamics is purely driven by dissipation and decoherence. Mathematically, they are formulated as completely positive trace preserving (CPTP) maps on the space of density matrices for the quantum walker on the graph. A microscopic derivation of the open quantum walks has been achieved in which it has been shown that all OQWs must include the possibility of remaining on the same site on the graph when the map is applied. In this work we extend the CPTP map to describe a lazy open quantum walk on a d-dimensional lattice. We study the asymptotic behaviour of this model and generalize an already existing central limit theorem for OQWs on the lattice to now include the lazy case.

Communications in Theoretical Physics

We investigate the evolution of a discrete-time one-dimensional quantum walk driven by a positiondependent coin. The rotation angle which depends upon the position of a quantum particle parameterizes the coin operator. For different values of the rotation angle, we observe that such a coin leads to a variety of probability distributions, e.g. localized, periodic, classical-like, semi-classicallike, and quantum-like. Further, we study the Shannon entropy associated with position space and coin space of a quantum particle and compare it with the case of the position-independent coin. We show that the entropy is smaller for most values of the rotation angle as compared to the case of the position-independent coin. We also study the effect of entanglement on the behavior of probability distribution and Shannon entropy of a quantum walk by considering two identical position-dependent entangled coins. We observe that in general, a quantum particle becomes more localized as compared to the case of the position-independent coin and hence the corresponding Shannon entropy is minimum. Our results show that position-dependent coin can be used as a controlling tool of quantum walks.

Quantum Information Processing, 2004

We survey the equations of continuous-time quantum walks on simple one-dimensional lattices, which include the finite and infinite lines and the finite cycle, and compare them with the classical continuous-time Markov chains. The focus of our expository article is on analyzing these processes using the Laplace transform on the stochastic recurrences. The resulting time evolution equations, classical versus quantum, are strikingly similar in form, although dissimilar in behavior. We also provide comparisons with analyses performed using spectral methods.

arXiv: Quantum Physics, 2009

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of up to two previous steps. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

Cornell University - arXiv, 2020

We propose an intermediate walk continuously connecting an open quantum random walk and a quantum walk with parameters M ∈ N controlling a decoherence effect; if M = 1, the walk coincides with an open quantum random walk, while M = ∞, the walk coincides with a quantum walk. We define a measure which recovers usual probability measures on Z for M = ∞ 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.

Quantum Information Processing, 2012

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers.

2010

We consider crossovers with respect to the weak convergence theorems from a discretetime quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete-and continuous-time random walks can be expressed as DTQWs in some limits. At first we generalize our previous study [Phys. Rev. A 81, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability p ∼ 1/n β can be evaluated, where n is the final time and 0 < β < 1. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter β shifts from 0 to 1 in both discrete-and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time n. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete-and continuous-time quantum and random walks.

Journal of Mathematical Sciences, 2020

We consider continuous unitary quantum walks of a single particle and of closed systems and discuss more realistic approaches to modeling quantum walks.

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogue of classical Markov chains. We explore the "quantum trajectory" point of view on these quantum random walks, that is, we show that measuring the position of the particle after each timestep gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Connections and differences with the already well-known quantum random walks, such as the Hadamard random walk, are established. We explore several examples and compute their limit behavior. We show that the typical behavior of Open Quantum Random Walks seems to be very different from Hadamardtype quantum random walks. Indeed, while being very quantum in their behavior, Open Quantum Random Walks tend to become more and more classical as time goes.

Physical Review A, 2015

Exploiting multi-dimensional quantum walks as feasible platforms for quantum computation and quantum simulation is attracting constantly growing attention from a broad experimental physics community. We propose a modification of the quantum walk scheme described in [C. Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)] that presents, in the considered regimes, a strong localization-like effect on the walker. We characterize it in terms of the parameters of a step-dependent qubit operation that acts on the coin space before any standard coin operation, showing that a proper choice can guarantee a non-negligible probability of finding the walker at the origin even for large times. We finally discuss possible experimental realizations of this model with the current state-of-the-art settings.