4,144 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
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
Quantum Fluctuations of Black Hole Geometry
By using the minisuperspace model for the interior metric ofstatic black
holes, we solve the Wheeler-DeWitt equation to study quantum mechanics of the
horizon geometry. Our basic idea is to introduce the gravitational mass and the
expansions of null rays as quantum operators. Then, the exact wave function is
found as a mass eigenstate, and the radius of the apparent horizon is
quantum-mechanically defined. In the evolution of the metric variables, the
wave function changes from a WKB solution giving the classical trajectories to
a tunneling solution. By virtue of the quantum fluctuations of the metric
evolution beyond the WKB approximation, we can observe a static black hole
state with the apparent horizon separating from the event horizon.Comment: 18 pages, DPNU-93-3
Spectrophotometric observations of comet P/Giacobini-Zinner
Spectroscopic observations of the Giacobini-Zinner comet were performed on March 20, June 20 and 21, September 11, and October 19, 1985. The September observations were performed at perihelion, exactly at the time of the International Cometary Explorer (ICE) encounter with the comet. The March and June observations were obtained with an intensified image dissector scanner (IIDS) on the 2.1-meter Kitt Peak telescope and the September and the October observations were obtained with a charge-coupled device (CCD) on the 4-meter Kitt Peak telescope. The nucleus spectra from these observations are presented
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
Entanglement measurement with discrete multiple coin quantum walks
Within a special multi-coin quantum walk scheme we analyze the effect of the
entanglement of the initial coin state. For states with a special entanglement
structure it is shown that this entanglement can be meausured with the mean
value of the walk, which depends on the i-concurrence of the initial coin
state. Further on the entanglement evolution is investigated and it is shown
that the symmetry of the probability distribution is reflected by the symmetry
of the entanglement distribution.Comment: 9 pages, IOP styl
Localization of Two-Dimensional Quantum Walks
The Grover walk, which is related to the Grover's search algorithm on a
quantum computer, is one of the typical discrete time quantum walks. However, a
localization of the two-dimensional Grover walk starting from a fixed point is
striking different from other types of quantum walks. The present paper
explains the reason why the walker who moves according to the degree-four
Grover's operator can remain at the starting point with a high probability. It
is shown that the key factor for the localization is due to the degeneration of
eigenvalues of the time evolution operator. In fact, the global time evolution
of the quantum walk on a large lattice is mainly determined by the degree of
degeneration. The dependence of the localization on the initial state is also
considered by calculating the wave function analytically.Comment: 21 pages RevTeX, 4 figures ep
Site-bond representation and self-duality for totalistic probabilistic cellular automata
We study the one-dimensional two-state totalistic probabilistic cellular
automata (TPCA) having an absorbing state with long-range interactions, which
can be considered as a natural extension of the Domany-Kinzel model. We
establish the conditions for existence of a site-bond representation and
self-dual property. Moreover we present an expression of a set-to-set
connectedness between two sets, a matrix expression for a condition of the
self-duality, and a convergence theorem for the TPCA.Comment: 11 pages, minor 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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