2,936 research outputs found
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
Quantum Walks on the Line with Phase Parameters
In this paper, a study on discrete-time coined quantum walks on the line is
presented. Clear mathematical foundations are still lacking for this quantum
walk model. As a step towards this objective, the following question is being
addressed: {\it Given a graph, what is the probability that a quantum walk
arrives at a given vertex after some number of steps?} This is a very natural
question, and for random walks it can be answered by several different
combinatorial arguments. For quantum walks this is a highly non-trivial task.
Furthermore, this was only achieved before for one specific coin operator
(Hadamard operator) for walks on the line. Even considering only walks on
lines, generalizing these computations to a general SU(2) coin operator is a
complex task. The main contribution is a closed-form formula for the amplitudes
of the state of the walk (which includes the question above) for a general
symmetric SU(2) operator for walks on the line. To this end, a coin operator
with parameters that alters the phase of the state of the walk is defined.
Then, closed-form solutions are computed by means of Fourier analysis and
asymptotic approximation methods. We also present some basic properties of the
walk which can be deducted using weak convergence theorems for quantum walks.
In particular, the support of the induced probability distribution of the walk
is calculated. Then, it is shown how changing the parameters in the coin
operator affects the resulting probability distribution.Comment: In v2 a small typo was fixed. The exponent in the definition of N_j
in Theorem 3 was changed from -1/2 to 1. 20 pages, 3 figures. Presented at
10th Asian Conference on Quantum Information Science (AQIS'10). Tokyo, Japan.
August 27-31, 201
Crossovers induced by discrete-time quantum walks
We consider crossovers with respect to the weak convergence theorems from a
discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk
(CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs
in some limit. At first we generalize our previous study [Phys. Rev. A
\textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show
that the position measurements per each step with probability can be evaluated, where is the final time and . We
also give a corresponding continuous-time case. As a consequence, crossovers
from the diffusive spreading (random walk) to the ballistic spreading (quantum
walk) can be seen as the parameter shifts from 0 to 1 in both discrete-
and continuous-time cases of the weak convergence theorems. Secondly, we
introduce a new class of the DTQW, in which the absolute value of the diagonal
parts of the quantum coin is proportional to a power of the inverse of the
final time . This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW
is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence
theorem for the FTD-DTQW which shows a variety of spreading properties.
Finally, we consider the FTD-DTQW with periodic position measurements. This
weak convergence theorem gives a phase diagram which maps sufficiently
long-time behaviors of the discrete- and continuous-time quantum and random
walks.Comment: 14 pages, 1 figur
Probability distributions and weak limit theorems of quaternionic quantum walks in one dimension
The discrete-time quantum walk (QW) is determined by a unitary matrix whose
component is complex number. Konno (2015) extended the QW to a walk whose
component is quaternion.We call this model quaternionic quantum walk (QQW). The
probability distribution of a class of QQWs is the same as that of the QW. On
the other hand, a numerical simulation suggests that the probability
distribution of a QQW is different from the QW. In this paper, we clarify the
difference between the QQW and the QW by weak limit theorems for a class of
QQWs.Comment: 11 pages, 2 figures, Interdisciplinary Information Sciences (in
press
Quantum walks and orbital states of a Weyl particle
The time-evolution equation of a one-dimensional quantum walker is exactly
mapped to the three-dimensional Weyl equation for a zero-mass particle with
spin 1/2, in which each wave number k of walker's wave function is mapped to a
point \vec{q}(k) in the three-dimensional momentum space and \vec{q}(k) makes a
planar orbit as k changes its value in [-\pi, \pi). The integration over k
providing the real-space wave function for a quantum walker corresponds to
considering an orbital state of a Weyl particle, which is defined as a
superposition (curvilinear integration) of the energy-momentum eigenstates of a
free Weyl equation along the orbit. Konno's novel distribution function of
quantum-walker's pseudo-velocities in the long-time limit is fully controlled
by the shape of the orbit and how the orbit is embedded in the
three-dimensional momentum space. The family of orbital states can be regarded
as a geometrical representation of the unitary group U(2) and the present study
will propose a new group-theoretical point of view for quantum-walk problems.Comment: REVTeX4, 9 pages, 1 figure, v2: Minor corrections made for
publication in Phys.Rev.
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