445 research outputs found
Some Remarks on the Matching Polynomial and Its Zeros
The matching polynomial (also called reference and acyclic
polynomial) was discovered in chemistry, physics and mathematics
at least six times. We demonstrate that the matching polynomial
of a bipartite graph coincides with the rook polynomial of a
certain board. The basic notions of rook theory17 are described. It
is also shown that the matching polynomial cannot always discriminate
between planar isospectral molecules
Some Remarks on the Matching Polynomial and Its Zeros
The matching polynomial (also called reference and acyclic
polynomial) was discovered in chemistry, physics and mathematics
at least six times. We demonstrate that the matching polynomial
of a bipartite graph coincides with the rook polynomial of a
certain board. The basic notions of rook theory17 are described. It
is also shown that the matching polynomial cannot always discriminate
between planar isospectral molecules
A simple encoding of a quantum circuit amplitude as a matrix permanent
A simple construction is presented which allows computing the transition
amplitude of a quantum circuit to be encoded as computing the permanent of a
matrix which is of size proportional to the number of quantum gates in the
circuit. This opens up some interesting classical monte-carlo algorithms for
approximating quantum circuits.Comment: 6 figure
A general algorithm for manipulating non-linear and linear entanglement witnesses by using exact convex optimization
A generic algorithm is developed to reduce the problem of obtaining linear
and nonlinear entanglement witnesses of a given quantum system, to convex
optimization problem. This approach is completely general and can be applied
for the entanglement detection of any N-partite quantum system. For this
purpose, a map from convex space of separable density matrices to a convex
region called feasible region is defined, where by using exact convex
optimization method, the linear entanglement witnesses can be obtained from
polygonal shape feasible regions, while for curved shape feasible regions,
envelope of the family of linear entanglement witnesses can be considered as
nonlinear entanglement witnesses. This method proposes a new methodological
framework within which most of previous EWs can be studied. To conclude and in
order to demonstrate the capability of the proposed approach, besides providing
some nonlinear witnesses for entanglement detection of density matrices in
unextendible product bases, W-states, and GHZ with W-states, some further
examples of three qubits systems and their classification and entanglement
detection are included. Also it is explained how one can manipulate most of the
non-decomposable linear and nonlinear three qubits entanglement witnesses
appearing in some of the papers published by us and other authors, by the
method proposed in this paper. Keywords: non-linear and linear entanglement
witnesses, convex optimization. PACS number(s): 03.67.Mn, 03.65.UdComment: 37 page
Pretty good state transfer in qubit chains-The Heisenberg Hamiltonian
Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission of a qubit state from one node of the network to another, with fidelity arbitrarily close to 1. We prove that in a Heisenberg chain with n qubits, there is pretty good state transfer between the nodes at the jth and (n − j + 1)th positions if n is a power of 2. Moreover, this condition is also necessary for j = 1. We obtain this result by applying a theorem due to Kronecker about Diophantine approximations, together with techniques from algebraic graph theory
Unitary designs and codes
A unitary design is a collection of unitary matrices that approximate the
entire unitary group, much like a spherical design approximates the entire unit
sphere. In this paper, we use irreducible representations of the unitary group
to find a general lower bound on the size of a unitary t-design in U(d), for
any d and t. We also introduce the notion of a unitary code - a subset of U(d)
in which the trace inner product of any pair of matrices is restricted to only
a small number of distinct values - and give an upper bound for the size of a
code of degree s in U(d) for any d and s. These bounds can be strengthened when
the particular inner product values that occur in the code or design are known.
Finally, we describe some constructions of designs: we give an upper bound on
the size of the smallest weighted unitary t-design in U(d), and we catalogue
some t-designs that arise from finite groups.Comment: 25 pages, no figure
On almost distance-regular graphs
Distance-regular graphs are a key concept in Algebraic Combinatorics and have
given rise to several generalizations, such as association schemes. Motivated
by spectral and other algebraic characterizations of distance-regular graphs,
we study `almost distance-regular graphs'. We use this name informally for
graphs that share some regularity properties that are related to distance in
the graph. For example, a known characterization of a distance-regular graph is
the invariance of the number of walks of given length between vertices at a
given distance, while a graph is called walk-regular if the number of closed
walks of given length rooted at any given vertex is a constant. One of the
concepts studied here is a generalization of both distance-regularity and
walk-regularity called -walk-regularity. Another studied concept is that of
-partial distance-regularity or, informally, distance-regularity up to
distance . Using eigenvalues of graphs and the predistance polynomials, we
discuss and relate these and other concepts of almost distance-regularity, such
as their common generalization of -walk-regularity. We introduce the
concepts of punctual distance-regularity and punctual walk-regularity as a
fundament upon which almost distance-regular graphs are built. We provide
examples that are mostly taken from the Foster census, a collection of
symmetric cubic graphs. Two problems are posed that are related to the question
of when almost distance-regular becomes whole distance-regular. We also give
several characterizations of punctually distance-regular graphs that are
generalizations of the spectral excess theorem
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