828 research outputs found
On the digraph of a unitary matrix
Given a matrix M of size n, a digraph D on n vertices is said to be the
digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is
an arc of D. We give a necessary condition, called strong quadrangularity, for
a digraph to be the digraph of a unitary matrix. With the use of such a
condition, we show that a line digraph, LD, is the digraph of a unitary matrix
if and only if D is Eulerian. It follows that, if D is strongly connected and
LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with
some elementary observations. Among the motivations of this paper are coined
quantum random walks, and, more generally, discrete quantum evolution on
digraphs.Comment: 6 page
On the Cayley digraphs that are patterns of unitary matrices
A digraph D is the pattern of a matrix M when D has an arc ij if and only if
the ij-th entry of M is nonzero. Study the relationship between unitary
matrices and their patterns is motivated by works in quantum chaology and
quantum computation. In this note, we prove that if a Cayley digraph is a line
digraph then it is the pattern of a unitary matrix. We prove that for any
finite group with two generators there exists a set of generators such that the
Cayley digraph with respect to such a set is a line digraph and hence the
pattern of a unitary matrix
Regular quantum graphs
We introduce the concept of regular quantum graphs and construct connected
quantum graphs with discrete symmetries. The method is based on a decomposition
of the quantum propagator in terms of permutation matrices which control the
way incoming and outgoing channels at vertex scattering processes are
connected. Symmetry properties of the quantum graph as well as its spectral
statistics depend on the particular choice of permutation matrices, also called
connectivity matrices, and can now be easily controlled. The method may find
applications in the study of quantum random walks networks and may also prove
to be useful in analysing universality in spectral statistics.Comment: 12 pages, 3 figure
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