2,944 research outputs found
Limit theorem for a time-dependent coined quantum walk on the line
We study time-dependent discrete-time quantum walks on the one-dimensional
lattice. We compute the limit distribution of a two-period quantum walk defined
by two orthogonal matrices. For the symmetric case, the distribution is
determined by one of two matrices. Moreover, limit theorems for two special
cases are presented
A quantum walk with a delocalized initial state: contribution from a coin-flip operator
A unit evolution step of discrete-time quantum walks is determined by both a
coin-flip operator and a position-shift operator. The behavior of quantum
walkers after many steps delicately depends on the coin-flip operator and an
initial condition of the walk. To get the behavior, a lot of long-time limit
distributions for the quantum walks starting with a localized initial state
have been derived. In the present paper, we compute limit distributions of a
2-state quantum walk with a delocalized initial state, not a localized initial
state, and discuss how the walker depends on the coin-flip operator. The
initial state induced from the Fourier series expansion, which is called the
delocalized initial state in this paper, provides different
limit density functions from the ones of the quantum walk with a localized
initial state.Comment: International Journal of Quantum Information, Vol.11, No.5, 1350053
(2013
Subcritical behavior in the alternating supercritical Domany-Kinzel dynamics
Cellular automata are widely used to model real-world dynamics. We show using
the Domany-Kinzel probabilistic cellular automata that alternating two
supercritical dynamics can result in subcritical dynamics in which the
population dies out. The analysis of the original and reduced models reveals
generality of this paradoxical behavior, which suggests that autonomous or
man-made periodic or random environmental changes can cause extinction in
otherwise safe population dynamics. Our model also realizes another scenario
for the Parrondo's paradox to occur, namely, spatial extensions.Comment: 8 figure
Wigner formula of rotation matrices and quantum walks
Quantization of a random-walk model is performed by giving a qudit (a
multi-component wave function) to a walker at site and by introducing a quantum
coin, which is a matrix representation of a unitary transformation. In quantum
walks, the qudit of walker is mixed according to the quantum coin at each time
step, when the walker hops to other sites. As special cases of the quantum
walks driven by high-dimensional quantum coins generally studied by Brun,
Carteret, and Ambainis, we study the models obtained by choosing rotation as
the unitary transformation, whose matrix representations determine quantum
coins. We show that Wigner's -dimensional unitary representations of
rotations with half-integers 's are useful to analyze the probability laws
of quantum walks. For any value of half-integer , convergence of all moments
of walker's pseudovelocity in the long-time limit is proved. It is generally
shown for the present models that, if is even, the probability measure
of limit distribution is given by a superposition of terms of scaled
Konno's density functions, and if is odd, it is a superposition of
terms of scaled Konno's density functions and a Dirac's delta function at the
origin. For the two-, three-, and four-component models, the probability
densities of limit distributions are explicitly calculated and their dependence
on the parameters of quantum coins and on the initial qudit of walker is
completely determined. Comparison with computer simulation results is also
shown.Comment: v2: REVTeX4, 15 pages, 4 figure
Localization of the Grover walks on spidernets and free Meixner laws
A spidernet is a graph obtained by adding large cycles to an almost regular
tree and considered as an example having intermediate properties of lattices
and trees in the study of discrete-time quantum walks on graphs. We introduce
the Grover walk on a spidernet and its one-dimensional reduction. We derive an
integral representation of the -step transition amplitude in terms of the
free Meixner law which appears as the spectral distribution. As an application
we determine the class of spidernets which exhibit localization. Our method is
based on quantum probabilistic spectral analysis of graphs.Comment: 32 page
A remark on zeta functions of finite graphs via quantum walks
From the viewpoint of quantum walks, the Ihara zeta function of a finite
graph can be said to be closely related to its evolution matrix. In this note
we introduce another kind of zeta function of a graph, which is closely related
to, as to say, the square of the evolution matrix of a quantum walk. Then we
give to such a function two types of determinant expressions and derive from it
some geometric properties of a finite graph. As an application, we illustrate
the distribution of poles of this function comparing with those of the usual
Ihara zeta function.Comment: 14 pages, 1 figur
Continuous-time quantum walk on integer lattices and homogeneous trees
This paper is concerned with the continuous-time quantum walk on Z, Z^d, and
infinite homogeneous trees. By using the generating function method, we compute
the limit of the average probability distribution for the general isotropic
walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous
trees. In addition, we compute the asymptotic approximation for the probability
of the return to zero at time t in all these cases.Comment: The journal version (save for formatting); 19 page
Breakdown of an Electric-Field Driven System: a Mapping to a Quantum Walk
Quantum transport properties of electron systems driven by strong electric
fields are studied by mapping the Landau-Zener transition dynamics to a quantum
walk on a semi-infinite one-dimensional lattice with a reflecting boundary,
where the sites correspond to energy levels and the boundary the ground state.
Quantum interference induces a distribution localized around the ground state,
and when the electric field is strengthened, a delocalization transition occurs
describing breakdown of the original electron system.Comment: 4 pages, 3 figures, Journal-ref adde
Absorption problems for quantum walks in one dimension
This paper treats absorption problems for the one-dimensional quantum walk
determined by a 2 times 2 unitary matrix U on a state space {0,1,...,N} where N
is finite or infinite by using a new path integral approach based on an
orthonormal basis P, Q, R and S of the vector space of complex 2 times 2
matrices. Our method studied here is a natural extension of the approach in the
classical random walk.Comment: 15 pages, small corrections, journal reference adde
Continuous deformations of the Grover walk preserving localization
The three-state Grover walk on a line exhibits the localization effect
characterized by a non-vanishing probability of the particle to stay at the
origin. We present two continuous deformations of the Grover walk which
preserve its localization nature. The resulting quantum walks differ in the
rate at which they spread through the lattice. The velocities of the left and
right-traveling probability peaks are given by the maximum of the group
velocity. We find the explicit form of peak velocities in dependence on the
coin parameter. Our results show that localization of the quantum walk is not a
singular property of an isolated coin operator but can be found for entire
families of coins
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