800 research outputs found
Survival of classical and quantum particles in the presence of traps
We present a detailed comparison of the motion of a classical and of a
quantum particle in the presence of trapping sites, within the framework of
continuous-time classical and quantum random walk. The main emphasis is on the
qualitative differences in the temporal behavior of the survival probabilities
of both kinds of particles. As a general rule, static traps are far less
efficient to absorb quantum particles than classical ones. Several lattice
geometries are successively considered: an infinite chain with a single trap, a
finite ring with a single trap, a finite ring with several traps, and an
infinite chain and a higher-dimensional lattice with a random distribution of
traps with a given density. For the latter disordered systems, the classical
and the quantum survival probabilities obey a stretched exponential asymptotic
decay, albeit with different exponents. These results confirm earlier
predictions, and the corresponding amplitudes are evaluated. In the
one-dimensional geometry of the infinite chain, we obtain a full analytical
prediction for the amplitude of the quantum problem, including its dependence
on the trap density and strength.Comment: 35 pages, 10 figures, 2 tables. Minor update
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
Recurrence for discrete time unitary evolutions
We consider quantum dynamical systems specified by a unitary operator U and
an initial state vector \phi. In each step the unitary is followed by a
projective measurement checking whether the system has returned to the initial
state. We call the system recurrent if this eventually happens with probability
one. We show that recurrence is equivalent to the absence of an absolutely
continuous part from the spectral measure of U with respect to \phi. We also
show that in the recurrent case the expected first return time is an integer or
infinite, for which we give a topological interpretation. A key role in our
theory is played by the first arrival amplitudes, which turn out to be the
(complex conjugated) Taylor coefficients of the Schur function of the spectral
measure. On the one hand, this provides a direct dynamical interpretation of
these coefficients; on the other hand it links our definition of first return
times to a large body of mathematical literature.Comment: 27 pages, 5 figures, typos correcte
Stationary measure for two-state space-inhomogeneous quantum walk in one dimension
We consider the two-state space-inhomogeneous coined quantum walk (QW) in one
dimension. For a general setting, we obtain the stationary measure of the QW by
solving the eigenvalue problem. As a corollary, stationary measures of the
multi-defect model and space-homogeneous QW are derived. The former is a
generalization of the previous works on one-defect model and the latter is a
generalization of the result given by Konno and Takei (2015).Comment: 15 pages, revised version, Yokohama Mathematical Journal (in press
Complete classification of trapping coins for quantum walks on the two-dimensional square lattice
One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, although the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four-state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four-dimensional coin Hilbert space. The well-known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such an escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the two-dimensional square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or nonexistence of the escaping state and the area covered by the spreading wave packet
- …