3 research outputs found
Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm
In papers\cite{js,jsa}, the amplitudes of continuous-time quantum walk on
graphs possessing quantum decomposition (QD graphs) have been calculated by a
new method based on spectral distribution associated to their adjacency matrix.
Here in this paper, it is shown that the continuous-time quantum walk on any
arbitrary graph can be investigated by spectral distribution method, simply by
using Krylov subspace-Lanczos algorithm to generate orthonormal bases of
Hilbert space of quantum walk isomorphic to orthogonal polynomials. Also new
type of graphs possessing generalized quantum decomposition have been
introduced, where this is achieved simply by relaxing some of the constrains
imposed on QD graphs and it is shown that both in QD and GQD graphs, the unit
vectors of strata are identical with the orthonormal basis produced by Lanczos
algorithm. Moreover, it is shown that probability amplitude of observing walk
at a given vertex is proportional to its coefficient in the corresponding unit
vector of its stratum, and it can be written in terms of the amplitude of its
stratum. Finally the capability of Lanczos-based algorithm for evaluation of
walk on arbitrary graphs (GQD or non-QD types), has been tested by calculating
the probability amplitudes of quantum walk on some interesting finite
(infinite) graph of GQD type and finite (infinite) path graph of non-GQD type,
where the asymptotic behavior of the probability amplitudes at infinite limit
of number of vertices, are in agreement with those of central limit theorem of
Ref.\cite{nko}.Comment: 29 pages, 4 figure
Evaluation of effective resistances in pseudo-distance-regular resistor networks
In Refs.[1] and [2], calculation of effective resistances on distance-regular
networks was investigated, where in the first paper, the calculation was based
on the stratification of the network and Stieltjes function associated with the
network, whereas in the latter one a recursive formula for effective
resistances was given based on the Christoffel-Darboux identity. In this paper,
evaluation of effective resistances on more general networks called
pseudo-distance-regular networks [21] or QD type networks \cite{obata} is
investigated, where we use the stratification of these networks and show that
the effective resistances between a given node such as and all of the
nodes belonging to the same stratum with respect to
(, belonging to the -th stratum with respect
to the ) are the same. Then, based on the spectral techniques, an
analytical formula for effective resistances such that
(those nodes , of
the network such that the network is symmetric with respect to them) is given
in terms of the first and second orthogonal polynomials associated with the
network, where is the pseudo-inverse of the Laplacian of the network.
From the fact that in distance-regular networks,
is satisfied for all nodes
of the network, the effective resistances
for ( is diameter of the network which
is the same as the number of strata) are calculated directly, by using the
given formula.Comment: 30 pages, 7 figure
Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm
In papers [Jafarizadehn and Salimi, Ann. Phys. 322, 1005 (2007) and J. Phys. A: Math. Gen. 39, 13295 (2006)], the amplitudes of continuous-time quantum
walk (CTQW) on graphs possessing quantum decomposition (QD graphs)
have been calculated by a new method based on spectral
distribution associated with their adjacency matrix. Here in this
paper, it is shown that the CTQW on any arbitrary graph can be
investigated by spectral analysis method, simply by using Krylov
subspace-Lanczos algorithm to generate orthonormal bases of
Hilbert space of quantum walk isomorphic to orthogonal
polynomials. Also new type of graphs possessing generalized
quantum decomposition (GQD) have been introduced, where this is
achieved simply by relaxing some of the constrains imposed on QD
graphs and it is shown that both in QD and GQD graphs, the unit
vectors of strata are identical with the orthonormal basis
produced by Lanczos algorithm. Moreover, it is shown that
probability amplitude of observing the walk at a given vertex is
proportional to its coefficient in the corresponding unit vector
of its stratum, and it can be written in terms of the amplitude of
its stratum. The capability of Lanczos-based algorithm for
evaluation of CTQW on graphs (GQD or non-QD types), has been
tested by calculating the probability amplitudes of quantum walk
on some interesting finite (infinite) graph of GQD type and finite
(infinite) path graph of non-GQD type, where the asymptotic
behavior of the probability amplitudes at the limit of the large
number of vertices, are in agreement with those of central limit
theorem of [Phys. Rev. E 72, 026113 (2005)]. At the end, some applications of the
method such as implementation of quantum search algorithms,
calculating the resistance between two nodes in regular networks
and applications in solid state and condensed matter physics, have
been discussed, where in all of them, the Lanczos algorithm,
reduces the Hilbert space to some smaller subspaces and the
problem is investigated in the subspace with maximal dimension