648 research outputs found
Symmetry-based matrix factorization
AbstractWe present a method for factoring a given matrix M into a short product of sparse matrices, provided that M has a suitable “symmetry”. This sparse factorization represents a fast algorithm for the matrix–vector multiplication with M. The factorization method consists of two essential steps. First, a combinatorial search is used to compute a suitable symmetry of M in the form of a pair of group representations. Second, the group representations are decomposed stepwise, which yields factorized decomposition matrices and determines a sparse factorization of M. The focus of this article is the first step, finding the symmetries. All algorithms described have been implemented in the library AREP. We present examples for automatically generated sparse factorizations—and hence fast algorithms—for a class of matrices corresponding to digital signal processing transforms including the discrete Fourier, cosine, Hartley, and Haar transforms
Some quasitensor autoequivalences of Drinfeld doubles of finite groups
We report on two classes of autoequivalences of the category of
Yetter-Drinfeld modules over a finite group, or, equivalently the Drinfeld
center of the category of representations of a finite group. Both operations
are related to the -th power operation, with relatively prime to the
exponent of the group. One is defined more generally for the group-theoretical
fusion category defined by a finite group and an arbitrary subgroup, while the
other seems particular to the case of Yetter-Drinfeld modules. Both
autoequivalences preserve higher Frobenius-Schur indicators up to Galois
conjugation, and they preserve tensor products, although neither of them can in
general be endowed with the structure of a monoidal functor.Comment: 18 page
A Frobenius formula for the structure coefficients of double-class algebras of Gelfand pairs
We generalise some well known properties of irreducible characters of finite
groups to zonal spherical functions of Gelfand pairs. This leads to a Frobenius
formula for Gelfand pairs. For a given Gelfand pair, the structure coefficients
of its associated double-class algebra can be written in terms of zonal
spherical functions. This is a generalisation of the Frobenius formula which
writes the structure coefficients of the center of a finite group algebra in
terms of irreducible characters
Flexible quantum circuits using scalable continuous-variable cluster states
We show that measurement-based quantum computation on scalable
continuous-variable (CV) cluster states admits more quantum-circuit flexibility
and compactness than similar protocols for standard square-lattice CV cluster
states. This advantage is a direct result of the macronode structure of these
states---that is, a lattice structure in which each graph node actually
consists of several physical modes. These extra modes provide additional
measurement degrees of freedom at each graph location, which can be used to
manipulate the flow and processing of quantum information more robustly and
with additional flexibility that is not available on an ordinary lattice.Comment: (v2) consistent with published version; (v1) 11 pages (9 figures
Group projector generalization of dirac-heisenberg model
The general form of the operators commuting with the ground representation
(appearing in many physical problems within single particle approximation) of
the group is found. With help of the modified group projector technique, this
result is applied to the system of identical particles with spin independent
interaction, to derive the Dirac-Heisenberg hamiltonian and its effective space
for arbitrary orbital occupation numbers and arbitrary spin. This gives
transparent insight into the physical contents of this hamiltonian, showing
that formal generalizations with spin greater than 1/2 involve nontrivial
additional physical assumptions.Comment: 10 page
Symmetry reduction induced by anyon condensation: a tensor network approach
Topological ordered phases are related to changes in the properties of their
quasi-particle excitations (anyons). We study these relations in the framework
of projected entanglement pair states (\textsf{PEPS}) and show how condensing
and confining anyons reduces a local gauge symmetry to a global on-site
symmetry. We also study the action of this global symmetry over the
quasiparticle excitations. As a byproduct, we observe that this symmetry
reduction effect can be applied to one-dimensional systems as well, and brings
about appealing physical interpretations on the classification of phases with
symmetries using matrix product states (\textsf{MPS}). The case of
on-site symmetry is studied in detail.Comment: 21+5 pages, 15+3 figures. Introduction and conclusions enlarged,
references and figure added, minor typos corrected, appendix about dyons
adde
Synthesis and Optimization of Reversible Circuits - A Survey
Reversible logic circuits have been historically motivated by theoretical
research in low-power electronics as well as practical improvement of
bit-manipulation transforms in cryptography and computer graphics. Recently,
reversible circuits have attracted interest as components of quantum
algorithms, as well as in photonic and nano-computing technologies where some
switching devices offer no signal gain. Research in generating reversible logic
distinguishes between circuit synthesis, post-synthesis optimization, and
technology mapping. In this survey, we review algorithmic paradigms ---
search-based, cycle-based, transformation-based, and BDD-based --- as well as
specific algorithms for reversible synthesis, both exact and heuristic. We
conclude the survey by outlining key open challenges in synthesis of reversible
and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table
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