40,116 research outputs found
A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs
For any polynomial representation of the special linear group, the nodes of
the corresponding crystal may be indexed by semi-standard Young tableaux. Under
certain conditions, the standard Young tableaux occur, and do so with weight 0.
Standard Young tableaux also parametrize the vertices of dual equivalence
graphs. Motivated by the underlying representation theory, in this paper, we
explainthis connection by giving a combinatorial manifestation of Schur-Weyl
duality. In particular, we put a dual equivalence graph structure on the
0-weight space of certain crystal graphs, producing edges combinatorially from
the crystal edges. The construction can be expressed in terms of the local
characterizations given by Stembridge for crystal graphs and the author for
dual equivalence graphs.Comment: 9 pages, 6 figures To appear in DMTCS as part of the FPSAC 2008
conference proceeding
Non-commutative flux representation for loop quantum gravity
The Hilbert space of loop quantum gravity is usually described in terms of
cylindrical functionals of the gauge connection, the electric fluxes acting as
non-commuting derivation operators. It has long been believed that this
non-commutativity prevents a dual flux (or triad) representation of loop
quantum gravity to exist. We show here, instead, that such a representation can
be explicitly defined, by means of a non-commutative Fourier transform defined
on the loop gravity state space. In this dual representation, flux operators
act by *-multiplication and holonomy operators act by translation. We describe
the gauge invariant dual states and discuss their geometrical meaning. Finally,
we apply the construction to the simpler case of a U(1) gauge group and compare
the resulting flux representation with the triad representation used in loop
quantum cosmology.Comment: 12 pages, matches published versio
Gunrock: GPU Graph Analytics
For large-scale graph analytics on the GPU, the irregularity of data access
and control flow, and the complexity of programming GPUs, have presented two
significant challenges to developing a programmable high-performance graph
library. "Gunrock", our graph-processing system designed specifically for the
GPU, uses a high-level, bulk-synchronous, data-centric abstraction focused on
operations on a vertex or edge frontier. Gunrock achieves a balance between
performance and expressiveness by coupling high performance GPU computing
primitives and optimization strategies with a high-level programming model that
allows programmers to quickly develop new graph primitives with small code size
and minimal GPU programming knowledge. We characterize the performance of
various optimization strategies and evaluate Gunrock's overall performance on
different GPU architectures on a wide range of graph primitives that span from
traversal-based algorithms and ranking algorithms, to triangle counting and
bipartite-graph-based algorithms. The results show that on a single GPU,
Gunrock has on average at least an order of magnitude speedup over Boost and
PowerGraph, comparable performance to the fastest GPU hardwired primitives and
CPU shared-memory graph libraries such as Ligra and Galois, and better
performance than any other GPU high-level graph library.Comment: 52 pages, invited paper to ACM Transactions on Parallel Computing
(TOPC), an extended version of PPoPP'16 paper "Gunrock: A High-Performance
Graph Processing Library on the GPU
Parallelizing Quantum Circuits
We present a novel automated technique for parallelizing quantum circuits via
forward and backward translation to measurement-based quantum computing
patterns and analyze the trade off in terms of depth and space complexity. As a
result we distinguish a class of polynomial depth circuits that can be
parallelized to logarithmic depth while adding only polynomial many auxiliary
qubits. In particular, we provide for the first time a full characterization of
patterns with flow of arbitrary depth, based on the notion of influencing paths
and a simple rewriting system on the angles of the measurement. Our method
leads to insightful knowledge for constructing parallel circuits and as
applications, we demonstrate several constant and logarithmic depth circuits.
Furthermore, we prove a logarithmic separation in terms of quantum depth
between the quantum circuit model and the measurement-based model.Comment: 34 pages, 14 figures; depth complexity, measurement-based quantum
computing and parallel computin
Complexity of Graph State Preparation
The graph state formalism is a useful abstraction of entanglement. It is used
in some multipartite purification schemes and it adequately represents
universal resources for measurement-only quantum computation. We focus in this
paper on the complexity of graph state preparation. We consider the number of
ancillary qubits, the size of the primitive operators, and the duration of
preparation. For each lexicographic order over these parameters we give upper
and lower bounds for the complexity of graph state preparation. The first part
motivates our work and introduces basic notions and notations for the study of
graph states. Then we study some graph properties of graph states,
characterizing their minimal degree by local unitary transformations, we
propose an algorithm to reduce the degree of a graph state, and show the
relationship with Sutner sigma-game.
These properties are used in the last part, where algorithms and lower bounds
for each lexicographic order over the considered parameters are presented.Comment: 17 page
Graphical Verification of a Spatial Logic for the Graphical Verification of a Spatial Logic for the pi-calculus
The paper introduces a novel approach to the verification of spatial properties for finite [pi]-calculus specifications. The mechanism is based on a recently proposed graphical encoding for mobile calculi: Each process is mapped into a (ranked) graph, such that the denotation is fully abstract with respect to the usual structural congruence (i.e., two processes are equivalent exactly when the corresponding encodings yield the same graph). Spatial properties for reasoning about the behavior and the structure of pi-calculus processes are then expressed in a logic introduced by Caires, and they are verified on the graphical encoding of a process, rather than on its textual representation. More precisely, the graphical presentation allows for providing a simple and easy to implement verification algorithm based on the graphical encoding (returning true if and only if a given process verifies a given spatial formula)
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