757,205 research outputs found
Hiding solutions in random satisfiability problems: A statistical mechanics approach
A major problem in evaluating stochastic local search algorithms for
NP-complete problems is the need for a systematic generation of hard test
instances having previously known properties of the optimal solutions. On the
basis of statistical mechanics results, we propose random generators of hard
and satisfiable instances for the 3-satisfiability problem (3SAT). The design
of the hardest problem instances is based on the existence of a first order
ferromagnetic phase transition and the glassy nature of excited states. The
analytical predictions are corroborated by numerical results obtained from
complete as well as stochastic local algorithms.Comment: 5 pages, 4 figures, revised version to app. in PR
A Random Matrix Model of Adiabatic Quantum Computing
We present an analysis of the quantum adiabatic algorithm for solving hard
instances of 3-SAT (an NP-complete problem) in terms of Random Matrix Theory
(RMT). We determine the global regularity of the spectral fluctuations of the
instantaneous Hamiltonians encountered during the interpolation between the
starting Hamiltonians and the ones whose ground states encode the solutions to
the computational problems of interest. At each interpolation point, we
quantify the degree of regularity of the average spectral distribution via its
Brody parameter, a measure that distinguishes regular (i.e., Poissonian) from
chaotic (i.e., Wigner-type) distributions of normalized nearest-neighbor
spacings. We find that for hard problem instances, i.e., those having a
critical ratio of clauses to variables, the spectral fluctuations typically
become irregular across a contiguous region of the interpolation parameter,
while the spectrum is regular for easy instances. Within the hard region, RMT
may be applied to obtain a mathematical model of the probability of avoided
level crossings and concomitant failure rate of the adiabatic algorithm due to
non-adiabatic Landau-Zener type transitions. Our model predicts that if the
interpolation is performed at a uniform rate, the average failure rate of the
quantum adiabatic algorithm, when averaged over hard problem instances, scales
exponentially with increasing problem size.Comment: 9 pages, 7 figure
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