6,026 research outputs found
New practical algorithms for the approximate shortest lattice vector
We present a practical algorithm that given an LLL-reduced lattice basis of dimension n, runs in time O(n3(k=6)k=4+n4) and approximates the length of the shortest, non-zero lattice vector to within a factor (k=6)n=(2k). This result is based on reasonable heuristics. Compared to previous practical algorithms the new method reduces the proven approximation factor achievable in a given time to less than its fourthth root. We also present a sieve algorithm inspired by Ajtai, Kumar, Sivakumar [AKS01]
Quantum Annealing and Analog Quantum Computation
We review here the recent success in quantum annealing, i.e., optimization of
the cost or energy functions of complex systems utilizing quantum fluctuations.
The concept is introduced in successive steps through the studies of mapping of
such computationally hard problems to the classical spin glass problems. The
quantum spin glass problems arise with the introduction of quantum
fluctuations, and the annealing behavior of the systems as these fluctuations
are reduced slowly to zero. This provides a general framework for realizing
analog quantum computation.Comment: 22 pages, 7 figs (color online); new References Added. Reviews of
Modern Physics (in press
On the sphere-decoding algorithm I. Expected complexity
The problem of finding the least-squares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worst-case complexity of the integer least-squares problem, we study its expected complexity, averaged over the noise and over the lattice. For the "sphere decoding" algorithm of Fincke and Pohst, we find a closed-form expression for the expected complexity, both for the infinite and finite lattice. It is demonstrated in the second part of this paper that, for a wide range of signal-to-noise ratios (SNRs) and numbers of antennas, the expected complexity is polynomial, in fact, often roughly cubic. Since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial, this suggests that maximum-likelihood decoding, which was hitherto thought to be computationally intractable, can, in fact, be implemented in real time - a result with many practical implications
Generation of confinement and other nonperturbative effects by infrared gluonic degrees of freedom
Recent progress in understanding the emergence of confinement and other
nonperturbative effects in the strong interaction vacuum is reviewed. Special
emphasis is placed on the role of different types of collective infrared
gluonic degrees of freedom in this respect. After a survey of complementary
approaches, models of the QCD vacuum based on center vortices, Abelian magnetic
monopoles and topological charge lumps such as instantons, merons and calorons
are examined. Both the physical mechanisms governing these models as well as
recent lattice studies of the respective degrees of freedom are reviewed.Comment: 15 pages (14 if you use the original espcrc2.sty instead of the
hypertex-compatible local version), 4 figures containing 7 postscript files,
talk presented at Lattice2004(plenary), Fermilab, June 21-26, 2004. Replaced
version contains an additional reference and a corresponding commen
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