33,801 research outputs found
Scaling a unitary matrix
The iterative method of Sinkhorn allows, starting from an arbitrary real
matrix with non-negative entries, to find a so-called 'scaled matrix' which is
doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and
with all line sums equal to 1. We conjecture that a similar procedure exists,
which allows, starting from an arbitrary unitary matrix, to find a scaled
matrix which is unitary and has all line sums equal to 1. The existence of such
algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.Comment: A proof of the conjecture is now provided by Idel & Wolf
(http://arxiv.org/abs/1408.5728
Unitary-Matrix Integration on 2D Yang-Mills Action
Using the idea of Itzykson-Zuber integral, unitary-matrix integration of 2D
Yang-Mills action is presented. The uniqueness of the solution of heat equation
enables us to integrate out the unitary-matrix parts of hermite matrices and to
obtain the expression of integration over vectors, also in this case.Comment: 8 page
Unitary Matrix Models and Phase Transition
We study the unitary matrix model with a topological term. We call the
topological term the theta term. In the symmetric model there is the phase
transition between the strong and weak coupling regime at . If
the Wilson term is bigger than the theta term, there is the strong-weak
coupling phase transition at the same . On the other hand, if the
theta term is bigger than the Wilson term, there is only the strong coupling
regime. So the topological phase transition disappears in this case.Comment: 9 pages, LaTeX, Comments about the topological phase transition are
adde
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
- …