16 research outputs found
Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations
Quantum walks subject to decoherence generically suffer the loss of their
genuine quantum feature, a quadratically faster spreading compared to classical
random walks. This intuitive statement has been verified analytically for
certain models and is also supported by numerical studies of a variety of
examples. In this paper we analyze the long-time behavior of a particular class
of decoherent quantum walks, which, to the best of our knowledge, was only
studied at the level of numerical simulations before. We consider a local coin
operation which is randomly and independently chosen for each time step and
each lattice site and prove that, under rather mild conditions, this leads to
classical behavior: With the same scaling as needed for a classical diffusion
the position distribution converges to a Gaussian, which is independent of the
initial state. Our method is based on non-degenerate perturbation theory and
yields an explicit expression for the covariance matrix of the asymptotic
Gaussian in terms of the randomness parameters
Stationary measure for two-state space-inhomogeneous quantum walk in one dimension
We consider the two-state space-inhomogeneous coined quantum walk (QW) in one
dimension. For a general setting, we obtain the stationary measure of the QW by
solving the eigenvalue problem. As a corollary, stationary measures of the
multi-defect model and space-homogeneous QW are derived. The former is a
generalization of the previous works on one-defect model and the latter is a
generalization of the result given by Konno and Takei (2015).Comment: 15 pages, revised version, Yokohama Mathematical Journal (in press
Quantum and random walks as universal generators of probability distributions
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 loosing coherence, as well as
random walks that exhibit the characteristic fast propagation of a quantum
particle driven by a Hadamard coin.Comment: 8 pages, 2 figures; revised and enlarged versio
Discrete-time Quantum Walks in random artificial Gauge Fields
Discrete-time quantum walks (DTQWs) in random artificial electric and
gravitational fields are studied analytically and numerically. The analytical
computations are carried by a new method which allows a direct exact analytical
determination of the equations of motion obeyed by the average density
operator. It is proven that randomness induces decoherence and that the quantum
walks behave asymptotically like classical random walks. Asymptotic diffusion
coefficients are computed exactly. The continuous limit is also obtained and
discussed.Comment: 16 pages, 9 figures. Submitted to Physica
Central limit theorem for reducible and irreducible open quantum walks
In this work we aim at proving central limit theorems for open quantum walks
on . We study the case when there are various classes of vertices
in the network. Furthermore, we investigate two ways of distributing the vertex
classes in the network. First we assign the classes in a regular pattern.
Secondly, we assign each vertex a random class with a uniform distribution. For
each way of distributing vertex classes, we obtain an appropriate central limit
theorem, illustrated by numerical examples. These theorems may have application
in the study of complex systems in quantum biology and dissipative quantum
computation.Comment: 20 pages, 4 figure
Two-walker discrete-time quantum walks on the line with percolation
One goal in the quantum-walk research is the exploitation of the intrinsic
quantum nature of multiple walkers, in order to achieve the full computational
power of the model. Here we study the behaviour of two non-interacting
particles performing a quantum walk on the line when the possibility of lattice
imperfections, in the form of missing links, is considered. We investigate two
regimes, statical and dynamical percolation, that correspond to different time
scales for the imperfections evolution with respect to the quantum-walk one. By
studying the qualitative behaviour of three two-particle quantities for
different probabilities of having missing bonds, we argue that the chosen
symmetry under particle-exchange of the input state strongly affects the output
of the walk, even in noisy and highly non-ideal regimes. We provide evidence
against the possibility of gathering information about the walkers
indistinguishability from the observation of bunching phenomena in the output
distribution, in all those situations that require a comparison between
averaged quantities. Although the spread of the walk is not substantially
changed by the addition of a second particle, we show that the presence of
multiple walkers can be beneficial for a procedure to estimate the probability
of having a broken link.Comment: 16 pages, 9 figure