4,816 research outputs found
Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces
We study a special class of (real or complex) robust Hadamard matrices,
distinguished by the property that their projection onto a -dimensional
subspace forms a Hadamard matrix. It is shown that such a matrix of order
exists, if there exists a skew Hadamard matrix of this size. This is the case
for any even dimension , and for these dimensions we demonstrate that
a bistochastic matrix located at any ray of the Birkhoff polytope, (which
joins the center of this body with any permutation matrix), is unistochastic.
An explicit form of the corresponding unitary matrix , such that
, is determined by a robust Hadamard matrix. These unitary
matrices allow us to construct a family of orthogonal bases in the composed
Hilbert space of order . Each basis consists of vectors with the
same degree of entanglement and the constructed family interpolates between the
product basis and the maximally entangled basis.Comment: 17 page
A feasibility approach for constructing combinatorial designs of circulant type
In this work, we propose an optimization approach for constructing various
classes of circulant combinatorial designs that can be defined in terms of
autocorrelations. The problem is formulated as a so-called feasibility problem
having three sets, to which the Douglas-Rachford projection algorithm is
applied. The approach is illustrated on three different classes of circulant
combinatorial designs: circulant weighing matrices, D-optimal matrices, and
Hadamard matrices with two circulant cores. Furthermore, we explicitly
construct two new circulant weighing matrices, a and a
, whose existence was previously marked as unresolved in the most
recent version of Strassler's table
Introduction to topological quantum computation with non-Abelian anyons
Topological quantum computers promise a fault tolerant means to perform
quantum computation. Topological quantum computers use particles with exotic
exchange statistics called non-Abelian anyons, and the simplest anyon model
which allows for universal quantum computation by particle exchange or braiding
alone is the Fibonacci anyon model. One classically hard problem that can be
solved efficiently using quantum computation is finding the value of the Jones
polynomial of knots at roots of unity. We aim to provide a pedagogical,
self-contained, review of topological quantum computation with Fibonacci
anyons, from the braiding statistics and matrices to the layout of such a
computer and the compiling of braids to perform specific operations. Then we
use a simulation of a topological quantum computer to explicitly demonstrate a
quantum computation using Fibonacci anyons, evaluating the Jones polynomial of
a selection of simple knots. In addition to simulating a modular circuit-style
quantum algorithm, we also show how the magnitude of the Jones polynomial at
specific points could be obtained exactly using Fibonacci or Ising anyons. Such
an exact algorithm seems ideally suited for a proof of concept demonstration of
a topological quantum computer.Comment: 51 pages, 51 figure
Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant
A bistochastic matrix B of size N is called unistochastic if there exists a
unitary U such that B_ij=|U_{ij}|^{2} for i,j=1,...,N. The set U_3 of all
unistochastic matrices of order N=3 forms a proper subset of the Birkhoff
polytope, which contains all bistochastic (doubly stochastic) matrices. We
compute the volume of the set U_3 with respect to the flat (Lebesgue) measure
and analytically evaluate the mean entropy of an unistochastic matrix of this
order. We also analyze the Jarlskog invariant J, defined for any unitary matrix
of order three, and derive its probability distribution for the ensemble of
matrices distributed with respect to the Haar measure on U(3) and for the
ensemble which generates the flat measure on the set of unistochastic matrices.
For both measures the probability of finding |J| smaller than the value
observed for the CKM matrix, which describes the violation of the CP parity, is
shown to be small. Similar statistical reasoning may also be applied to the MNS
matrix, which plays role in describing the neutrino oscillations. Some
conjectures are made concerning analogous probability measures in the space of
unitary matrices in higher dimensions.Comment: 33 pages, 6 figures version 2 - misprints corrected, explicit
formulae for phases provide
Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
We give a scheme for interpreting shaded tangles as quantum programs, with
the property that isotopic tangles yield equivalent programs. We analyze many
known quantum programs in this way -- including entanglement manipulation and
error correction -- and in each case present a fully-topological formal
verification, yielding in several cases substantial new insight into how the
program works. We also use our methods to identify several new or generalized
procedures.Comment: In Proceedings QPL 2017, arXiv:1802.0973
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