2,273 research outputs found
Exact solutions for diluted spin glasses and optimization problems
We study the low temperature properties of p-spin glass models with finite
connectivity and of some optimization problems. Using a one-step functional
replica symmetry breaking Ansatz we can solve exactly the saddle-point
equations for graphs with uniform connectivity. The resulting ground state
energy is in perfect agreement with numerical simulations. For fluctuating
connectivity graphs, the same Ansatz can be used in a variational way: For
p-spin models (known as p-XOR-SAT in computer science) it provides the exact
configurational entropy together with the dynamical and static critical
connectivities (for p=3, \gamma_d=0.818 and \gamma_s=0.918 resp.), whereas for
hard optimization problems like 3-SAT or Bicoloring it provides new upper
bounds for their critical thresholds (\gamma_c^{var}=4.396 and
\gamma_c^{var}=2.149 resp.).Comment: 4 pages, 1 figure, accepted for publication in PR
Minimizing energy below the glass thresholds
Focusing on the optimization version of the random K-satisfiability problem,
the MAX-K-SAT problem, we study the performance of the finite energy version of
the Survey Propagation (SP) algorithm. We show that a simple (linear time)
backtrack decimation strategy is sufficient to reach configurations well below
the lower bound for the dynamic threshold energy and very close to the analytic
prediction for the optimal ground states. A comparative numerical study on one
of the most efficient local search procedures is also given.Comment: 12 pages, submitted to Phys. Rev. E, accepted for publicatio
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Ground state of the Bethe-lattice spin glass and running time of an exact optimization algorithm
We study the Ising spin glass on random graphs with fixed connectivity z and
with a Gaussian distribution of the couplings, with mean \mu and unit variance.
We compute exact ground states by using a sophisticated branch-and-cut method
for z=4,6 and system sizes up to N=1280 for different values of \mu. We locate
the spin-glass/ferromagnet phase transition at \mu = 0.77 +/- 0.02 (z=4) and
\mu = 0.56 +/- 0.02 (z=6). We also compute the energy and magnetization in the
Bethe-Peierls approximation with a stochastic method, and estimate the
magnitude of replica symmetry breaking corrections. Near the phase transition,
we observe a sharp change of the median running time of our implementation of
the algorithm, consistent with a change from a polynomial dependence on the
system size, deep in the ferromagnetic phase, to slower than polynomial in the
spin-glass phase.Comment: 10 pages, RevTex, 10 eps figures. Some changes in the tex
Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses
We develop a systematic cluster expansion for dilute systems in the highly
dilute phase. We first apply it to the calculation of the entropy of the
K-satisfiability problem in the satisfiable phase. We derive a series expansion
in the control parameter, the average connectivity, that is identical to the
one obtained by using the replica approach with a replica symmetric ({\sc rs})
{\it Ansatz}, when the order parameter is calculated via a perturbative
expansion in the control parameter. As a second application we compute the
free-energy of the Viana-Bray model in the paramagnetic phase. The cluster
expansion allows one to compute finite-size corrections in a simple manner and
these are particularly important in optimization problems. Importantly enough,
these calculations prove the exactness of the {\sc rs} {\it Ansatz} below the
percolation threshold and might require its revision between this and the
easy-to-hard transition.Comment: 21 pages, 7 figs, to appear in Phys. Rev.
First-order transitions and the performance of quantum algorithms in random optimization problems
We present a study of the phase diagram of a random optimization problem in
presence of quantum fluctuations. Our main result is the characterization of
the nature of the phase transition, which we find to be a first-order quantum
phase transition. We provide evidence that the gap vanishes exponentially with
the system size at the transition. This indicates that the Quantum Adiabatic
Algorithm requires a time growing exponentially with system size to find the
ground state of this problem.Comment: 4 pages, 4 figures; final version accepted on Phys.Rev.Let
The nature of the different zero-temperature phases in discrete two-dimensional spin glasses: Entropy, universality, chaos and cascades in the renormalization group flow
The properties of discrete two-dimensional spin glasses depend strongly on
the way the zero-temperature limit is taken. We discuss this phenomenon in the
context of the Migdal-Kadanoff renormalization group. We see, in particular,
how these properties are connected with the presence of a cascade of fixed
points in the renormalization group flow. Of particular interest are two
unstable fixed points that correspond to two different spin-glass phases at
zero temperature. We discuss how these phenomena are related with the presence
of entropy fluctuations and temperature chaos, and universality in this model.Comment: 14 pages, 5 figures, 2 table
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