2,079 research outputs found
Protected gates for topological quantum field theories
We study restrictions on locality-preserving unitary logical gates for
topological quantum codes in two spatial dimensions. A locality-preserving
operation is one which maps local operators to local operators --- for example,
a constant-depth quantum circuit of geometrically local gates, or evolution for
a constant time governed by a geometrically-local bounded-strength Hamiltonian.
Locality-preserving logical gates of topological codes are intrinsically fault
tolerant because spatially localized errors remain localized, and hence
sufficiently dilute errors remain correctable. By invoking general properties
of two-dimensional topological field theories, we find that the
locality-preserving logical gates are severely limited for codes which admit
non-abelian anyons; in particular, there are no locality-preserving logical
gates on the torus or the sphere with M punctures if the braiding of anyons is
computationally universal. Furthermore, for Ising anyons on the M-punctured
sphere, locality-preserving gates must be elements of the logical Pauli group.
We derive these results by relating logical gates of a topological code to
automorphisms of the Verlinde algebra of the corresponding anyon model, and by
requiring the logical gates to be compatible with basis changes in the logical
Hilbert space arising from local F-moves and the mapping class group.Comment: 50 pages, many figures, v3: updated to match published versio
A review of parallel finite element methods on the DAP
AbstractThis paper reviews the research work that has been done to implement the finite element method for solving partial differential equations on the ICL distributed array processor (DAP). A brief outline of the principle features of the method is given, followed by details of the novel techniques required for implementation on the highly parallel architecture. Various methods of solution of the finite element equations are discussed; both direct and iterative techniques are included. The current state-of-the-art favours the use of the preconditioned conjugate gradient method. Some suggestions for future research work on parallel finite element methods are made
Solution of partial differential equations on vector and parallel computers
The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed
Mixing multi-core CPUs and GPUs for scientific simulation software
Recent technological and economic developments have led to widespread availability of
multi-core CPUs and specialist accelerator processors such as graphical processing units
(GPUs). The accelerated computational performance possible from these devices can be very
high for some applications paradigms. Software languages and systems such as NVIDIA's
CUDA and Khronos consortium's open compute language (OpenCL) support a number of
individual parallel application programming paradigms. To scale up the performance of some
complex systems simulations, a hybrid of multi-core CPUs for coarse-grained parallelism and
very many core GPUs for data parallelism is necessary. We describe our use of hybrid applica-
tions using threading approaches and multi-core CPUs to control independent GPU devices.
We present speed-up data and discuss multi-threading software issues for the applications
level programmer and o er some suggested areas for language development and integration
between coarse-grained and ne-grained multi-thread systems. We discuss results from three
common simulation algorithmic areas including: partial di erential equations; graph cluster
metric calculations and random number generation. We report on programming experiences
and selected performance for these algorithms on: single and multiple GPUs; multi-core CPUs;
a CellBE; and using OpenCL. We discuss programmer usability issues and the outlook and
trends in multi-core programming for scienti c applications developers
Lattice gauge theories dynamical fermions and parallel computation
SIGLEAvailable from British Library Document Supply Centre- DSC:D71683/87 / BLDSC - British Library Document Supply CentreGBUnited Kingdo
Zonotopal algebra
A wealth of geometric and combinatorial properties of a given linear
endomorphism of is captured in the study of its associated zonotope
, and, by duality, its associated hyperplane arrangement .
This well-known line of study is particularly interesting in case n\eqbd\rank
X \ll N. We enhance this study to an algebraic level, and associate with
three algebraic structures, referred herein as {\it external, central, and
internal.} Each algebraic structure is given in terms of a pair of homogeneous
polynomial ideals in variables that are dual to each other: one encodes
properties of the arrangement , while the other encodes by duality
properties of the zonotope . The algebraic structures are defined purely
in terms of the combinatorial structure of , but are subsequently proved to
be equally obtainable by applying suitable algebro-analytic operations to
either of or . The theory is universal in the sense that it
requires no assumptions on the map (the only exception being that the
algebro-analytic operations on yield sought-for results only in case
is unimodular), and provides new tools that can be used in enumerative
combinatorics, graph theory, representation theory, polytope geometry, and
approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments
in the area since the article first appeared on the arXi
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