28,270 research outputs found
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
New Bounds for the Garden-Hose Model
We show new results about the garden-hose model. Our main results include
improved lower bounds based on non-deterministic communication complexity
(leading to the previously unknown bounds for Inner Product mod 2
and Disjointness), as well as an upper bound for the
Distributed Majority function (previously conjectured to have quadratic
complexity). We show an efficient simulation of formulae made of AND, OR, XOR
gates in the garden-hose model, which implies that lower bounds on the
garden-hose complexity of the order will be
hard to obtain for explicit functions. Furthermore we study a time-bounded
variant of the model, in which even modest savings in time can lead to
exponential lower bounds on the size of garden-hose protocols.Comment: In FSTTCS 201
Theory of quantum Loschmidt echoes
In this paper we review our recent work on the theoretical approach to
quantum Loschmidt echoes, i.e. various properties of the so called echo
dynamics -- the composition of forward and backward time evolutions generated
by two slightly different Hamiltonians, such as the state autocorrelation
function (fidelity) and the purity of a reduced density matrix traced over a
subsystem (purity fidelity). Our main theoretical result is a linear response
formalism, expressing the fidelity and purity fidelity in terms of integrated
time autocorrelation function of the generator of the perturbation.
Surprisingly, this relation predicts that the decay of fidelity is the slower
the faster the decay of correlations. In particular for a static
(time-independent) perturbation, and for non-ergodic and non-mixing dynamics
where asymptotic decay of correlations is absent, a qualitatively different and
faster decay of fidelity is predicted on a time scale 1/delta as opposed to
mixing dynamics where the fidelity is found to decay exponentially on a
time-scale 1/delta^2, where delta is a strength of perturbation. A detailed
discussion of a semi-classical regime of small effective values of Planck
constant is given where classical correlation functions can be used to predict
quantum fidelity decay. Note that the correct and intuitively expected
classical stability behavior is recovered in the classical limit, as the
perturbation and classical limits do not commute. The theoretical results are
demonstrated numerically for two models, the quantized kicked top and the
multi-level Jaynes Cummings model. Our method can for example be applied to the
stability analysis of quantum computation and quantum information processing.Comment: 29 pages, 11 figures ; Maribor 2002 proceeding
From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond
The discovery of connections between the distribution of energy levels of
heavy nuclei and spacings between prime numbers has been one of the most
surprising and fruitful observations in the twentieth century. The connection
between the two areas was first observed through Montgomery's work on the pair
correlation of zeros of the Riemann zeta function. As its generalizations and
consequences have motivated much of the following work, and to this day remains
one of the most important outstanding conjectures in the field, it occupies a
central role in our discussion below. We describe some of the many techniques
and results from the past sixty years, especially the important roles played by
numerical and experimental investigations, that led to the discovery of the
connections and progress towards understanding the behaviors. In our survey of
these two areas, we describe the common mathematics that explains the
remarkable universality. We conclude with some thoughts on what might lie ahead
in the pair correlation of zeros of the zeta function, and other similar
quantities.Comment: Version 1.1, 50 pages, 6 figures. To appear in "Open Problems in
Mathematics", Editors John Nash and Michael Th. Rassias. arXiv admin note:
text overlap with arXiv:0909.491
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