21,345 research outputs found
Certified lattice reduction
Quadratic form reduction and lattice reduction are fundamental tools in
computational number theory and in computer science, especially in
cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm
(so-called LLL) has been improved in many ways through the past decades and
remains one of the central methods used for reducing integral lattice basis. In
particular, its floating-point variants-where the rational arithmetic required
by Gram-Schmidt orthogonalization is replaced by floating-point arithmetic-are
now the fastest known. However, the systematic study of the reduction theory of
real quadratic forms or, more generally, of real lattices is not widely
represented in the literature. When the problem arises, the lattice is usually
replaced by an integral approximation of (a multiple of) the original lattice,
which is then reduced. While practically useful and proven in some special
cases, this method doesn't offer any guarantee of success in general. In this
work, we present an adaptive-precision version of a generalized LLL algorithm
that covers this case in all generality. In particular, we replace
floating-point arithmetic by Interval Arithmetic to certify the behavior of the
algorithm. We conclude by giving a typical application of the result in
algebraic number theory for the reduction of ideal lattices in number fields.Comment: 23 page
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
Multi-core computation of transfer matrices for strip lattices in the Potts model
The transfer-matrix technique is a convenient way for studying strip lattices
in the Potts model since the compu- tational costs depend just on the periodic
part of the lattice and not on the whole. However, even when the cost is
reduced, the transfer-matrix technique is still an NP-hard problem since the
time T(|V|, |E|) needed to compute the matrix grows ex- ponentially as a
function of the graph width. In this work, we present a parallel
transfer-matrix implementation that scales performance under multi-core
architectures. The construction of the matrix is based on several repetitions
of the deletion- contraction technique, allowing parallelism suitable to
multi-core machines. Our experimental results show that the multi-core
implementation achieves speedups of 3.7X with p = 4 processors and 5.7X with p
= 8. The efficiency of the implementation lies between 60% and 95%, achieving
the best balance of speedup and efficiency at p = 4 processors for actual
multi-core architectures. The algorithm also takes advantage of the lattice
symmetry, making the transfer matrix computation to run up to 2X faster than
its non-symmetric counterpart and use up to a quarter of the original space
Spectrum of the Dirac Operator and Multigrid Algorithm with Dynamical Staggered Fermions
Complete spectra of the staggered Dirac operator \Dirac are determined in
quenched four-dimensional gauge fields, and also in the presence of
dynamical fermions.
Periodic as well as antiperiodic boundary conditions are used.
An attempt is made to relate the performance of multigrid (MG) and conjugate
gradient (CG) algorithms for propagators with the distribution of the
eigenvalues of~\Dirac.
The convergence of the CG algorithm is determined only by the condition
number~ and by the lattice size.
Since~'s do not vary significantly when quarks become dynamic,
CG convergence in unquenched fields can be predicted from quenched
simulations.
On the other hand, MG convergence is not affected by~ but depends on
the spectrum in a more subtle way.Comment: 19 pages, 8 figures, HUB-IEP-94/12 and KL-TH 19/94; comes as a
uuencoded tar-compressed .ps-fil
Light dynamical fermions on the lattice: toward the chiral regime of QCD
Algorithmic and technical progress achieved over the last few years makes QCD
simulations with light dynamical quarks much faster than before. As a result
lattices with pions as light as 250--300 MeV can be simulated with the present
generation of computers. I review recent conceptual and numerical progress in
this field, with particular emphasis on results obtained and difficulties
encountered in simulations with significantly smaller quark masses with respect
to previous computations. I also attempt to compare physical results for pion
masses and decay constants available to date in the two-flavour theory with
expectations from chiral perturbation theory.Comment: Plenary talk given at XXIVth International Symposium on Lattice Field
Theory Lattice2006(plenary), Tucson, Arizona, 23-28 July 2006. Submitted to
PoS in October 200
Quantum information and statistical mechanics: an introduction to frontier
This is a short review on an interdisciplinary field of quantum information
science and statistical mechanics. We first give a pedagogical introduction to
the stabilizer formalism, which is an efficient way to describe an important
class of quantum states, the so-called stabilizer states, and quantum
operations on them. Furthermore, graph states, which are a class of stabilizer
states associated with graphs, and their applications for measurement-based
quantum computation are also mentioned. Based on the stabilizer formalism, we
review two interdisciplinary topics. One is the relation between quantum error
correction codes and spin glass models, which allows us to analyze the
performances of quantum error correction codes by using the knowledge about
phases in statistical models. The other is the relation between the stabilizer
formalism and partition functions of classical spin models, which provides new
quantum and classical algorithms to evaluate partition functions of classical
spin models.Comment: 15pages, 4 figures, to appear in Proceedings of 4th YSM-SPIP (Sendai,
14-16 December 2012
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