371 research outputs found

### A random walk approach to quantum algorithms

The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial: pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e., when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk, due to interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point on average than a classical walker, and this forms the basis of a quantum speed up that can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, with even a small quantum computer available, development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems

### Limit theorems for a localization model of 2-state quantum walks

We consider 2-state quantum walks (QWs) on the line, which are defined by two
matrices. One of the matrices operates the walk at only half-time. In the usual
QWs, localization does not occur at all. However, our walk can be localized
around the origin. In this paper, we present two limit theorems, that is, one
is a stationary distribution and the other is a convergence theorem in
distribution.Comment: International Journal of Quantum Information, Vol.9, No.3, pp.863-874
(2011

### Quantum walks on general graphs

Quantum walks, both discrete (coined) and continuous time, on a general graph
of N vertices with undirected edges are reviewed in some detail. The resource
requirements for implementing a quantum walk as a program on a quantum computer
are compared and found to be very similar for both discrete and continuous time
walks. The role of the oracle, and how it changes if more prior information
about the graph is available, is also discussed.Comment: 8 pages, v2: substantial rewrite improves clarity, corrects errors
and omissions; v3: removes major error in final section and integrates
remainder into other sections, figures remove

### 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

### Decoherence vs entanglement in coined quantum walks

Quantum versions of random walks on the line and cycle show a quadratic
improvement in their spreading rate and mixing times respectively. The addition
of decoherence to the quantum walk produces a more uniform distribution on the
line, and even faster mixing on the cycle by removing the need for
time-averaging to obtain a uniform distribution. We calculate numerically the
entanglement between the coin and the position of the quantum walker and show
that the optimal decoherence rates are such that all the entanglement is just
removed by the time the final measurement is made.Comment: 11 pages, 6 embedded eps figures; v2 improved layout and discussio

### Quantum discord and the power of one qubit

We use quantum discord to characterize the correlations present in the
quantum computational model DQC1, introduced by Knill and Laflamme [Phys. Rev.
Lett. 81, 5672 (1998)]. The model involves a collection of qubits in the
completely mixed state coupled to a single control qubit that has nonzero
purity. The initial state, operations, and measurements in the model all point
to a natural bipartite split between the control qubit and the mixed ones.
Although there is no entanglement between these two parts, we show that the
quantum discord across this split is nonzero for typical instances of the DQC1
ciruit. Nonzero values of discord indicate the presence of nonclassical
correlations. We propose quantum discord as figure of merit for characterizing
the resources present in this computational model.Comment: 4 Pages, 1 Figur

### Inertial effects in three dimensional spinodal decomposition of a symmetric binary fluid mixture: A lattice Boltzmann study

The late-stage demixing following spinodal decomposition of a
three-dimensional symmetric binary fluid mixture is studied numerically, using
a thermodynamicaly consistent lattice Boltzmann method. We combine results from
simulations with different numerical parameters to obtain an unprecendented
range of length and time scales when expressed in reduced physical units. Using
eight large (256^3) runs, the resulting composite graph of reduced domain size
l against reduced time t covers 1 < l < 10^5, 10 < t < 10^8. Our data is
consistent with the dynamical scaling hypothesis, that l(t) is a universal
scaling curve. We give the first detailed statistical analysis of fluid motion,
rather than just domain evolution, in simulations of this kind, and introduce
scaling plots for several quantities derived from the fluid velocity and
velocity gradient fields.Comment: 49 pages, latex, J. Fluid Mech. style, 48 embedded eps figs plus 6
colour jpegs for Fig 10 on p.2

### Interface Width and Bulk Stability: requirements for the simulation of Deeply Quenched Liquid-Gas Systems

Simulations of liquid-gas systems with extended interfaces are observed to
fail to give accurate results for two reasons: the interface can get ``stuck''
on the lattice or a density overshoot develops around the interface. In the
first case the bulk densities can take a range of values, dependent on the
initial conditions. In the second case inaccurate bulk densities are found. In
this communication we derive the minimum interface width required for the
accurate simulation of liquid gas systems with a diffuse interface. We
demonstrate this criterion for lattice Boltzmann simulations of a van der Waals
gas. When combining this criterion with predictions for the bulk stability we
can predict the parameter range that leads to stable and accurate simulation
results. This allows us to identify parameter ranges leading to high density
ratios of over 1000. This is despite the fact that lattice Boltzmann
simulations of liquid-gas systems were believed to be restricted to modest
density ratios of less than 20.Comment: 5 pages, 3 figure

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