10,740 research outputs found
Quantum walks with memory provided by recycled coins and a memory of the coin-flip history
Quantum walks have emerged as an interesting approach to quantum information processing, exhibiting many unique properties compared to the analogous classical random walk. Here we introduce a model for a discrete-time quantum walk with memory by endowing the walker with multiple recycled coins and using a physical memory function via a history dependent coin flip. By numerical simulation we observe several phenomena. First in one dimension, walkers with memory have persistent quantum ballistic speed up over classical walks just as found in previous studies of multicoined walks with trivial memory function. However, measurement of the multicoin state can dramatically shift the mean of the spatial distribution. Second, we consider spatial entanglement in a two-dimensional quantum walk with memory and find that memory destroys entanglement between the spatial dimensions, even when entangling coins are employed. Finally, we explore behavior in the presence of spatial randomness and find that in the time regime where single-coined walks localize, multicoined walks do not and in fact a memory function can speed up the walk relative to a multicoin walker with no memory. We explicitly show how to construct linear optics circuits implementing the walks, and discuss prospects for classical simulation. © 2013 American Physical Society
The Elephant Quantum Walk
We explore the impact of long-range memory on the properties of a family of
quantum walks in a one-dimensional lattice and discrete time, which can be
understood as the quantum version of the classical "Elephant Random Walk"
non-Markovian process. This Elephant Quantum Walk is robustly superballistic
with the standard deviation showing a constant exponent, , whatever the quantum coin operator, on which the diffusion coefficient is
dependent. On the one hand, this result indicates that contrarily to the
classical case, the degree of superdiffusivity in quantum non- Markovian
processes of this kind is mainly ruled by the extension of memory rather than
other microscopic parameters that explicitly define the process. On the other
hand, these parameters reflect on the diffusion coefficient.Comment: 4 figures, any comments is welcome. Accepted in PR
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Discrete-time quantum walk with two-step memory
We examine a discrete-time quantum walk with two-step memory for a particle on a one-
dimensional 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
Non-Markovian continuous-time quantum walks on lattices with dynamical noise
We address the dynamics of continuous-time quantum walks on one-dimensional
disordered lattices inducing dynamical noise in the system. Noise is described
as time-dependent fluctuations of the tunneling amplitudes between adjacent
sites, and attention is focused on non-Gaussian telegraph noise, going beyond
the usual assumption of fast Gaussian noise. We observe the emergence of two
different dynamical behaviors for the walker, corresponding to two opposite
noise regimes: slow noise (i.e. strong coupling with the environment) confines
the walker into few lattice nodes, while fast noise (weak coupling) induces a
transition between quantum and classical diffusion over the lattice. A phase
transition between the two dynamical regimes may be observed by tuning the
ratio between the autocorrelation time of the noise and the coupling between
the walker and the external environment generating the noise. We also address
the non-Markovianity of the quantum map by assessing its memory effects, as
well as evaluating the information backflow to the system. Our results suggest
that the non-Markovian character of the evolution is linked to the dynamical
behavior in the slow noise regime, and that fast noise induces a Markovian
dynamics for the walker.Comment: 10 pages, 8 figure
The QWalk Simulator of Quantum Walks
Several research groups are giving special attention to quantum walks
recently, because this research area have been used with success in the
development of new efficient quantum algorithms. A general simulator of quantum
walks is very important for the development of this area, since it allows the
researchers to focus on the mathematical and physical aspects of the research
instead of deviating the efforts to the implementation of specific numerical
simulations. In this paper we present QWalk, a quantum walk simulator for one-
and two-dimensional lattices. Finite two-dimensional lattices with generic
topologies can be used. Decoherence can be simulated by performing measurements
or by breaking links of the lattice. We use examples to explain the usage of
the software and to show some recent results of the literature that are easily
reproduced by the simulator.Comment: 21 pages, 11 figures. Accepted in Computer Physics Communications.
Simulator can be downloaded from http://qubit.lncc.br/qwal
Quantum walks with random phase shifts
We investigate quantum walks in multiple dimensions with different quantum
coins. We augment the model by assuming that at each step the amplitudes of the
coin state are multiplied by random phases. This model enables us to study in
detail the role of decoherence in quantum walks and to investigate the
quantum-to-classical transition. We also provide classical analogues of the
quantum random walks studied. Interestingly enough, it turns out that the
classical counterparts of some quantum random walks are classical random walks
with a memory and biased coin. In addition random phase shifts "simplify" the
dynamics (the cross interference terms of different paths vanish on average)
and enable us to give a compact formula for the dispersion of such walks.Comment: to appear in Phys. Rev. A (10 pages, 5 figures
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