34,544 research outputs found
Single-Step Quantum Search Using Problem Structure
The structure of satisfiability problems is used to improve search algorithms
for quantum computers and reduce their required coherence times by using only a
single coherent evaluation of problem properties. The structure of random k-SAT
allows determining the asymptotic average behavior of these algorithms, showing
they improve on quantum algorithms, such as amplitude amplification, that
ignore detailed problem structure but remain exponential for hard problem
instances. Compared to good classical methods, the algorithm performs better,
on average, for weakly and highly constrained problems but worse for hard
cases. The analytic techniques introduced here also apply to other quantum
algorithms, supplementing the limited evaluation possible with classical
simulations and showing how quantum computing can use ensemble properties of NP
search problems.Comment: 39 pages, 12 figures. Revision describes further improvement with
multiple steps (section 7). See also
http://www.parc.xerox.com/dynamics/www/quantum.htm
The 3-SAT problem with large number of clauses in -replica symmetry breaking scheme
In this paper we analyze the structure of the UNSAT-phase of the
overconstrained 3-SAT model by studying the low temperature phase of the
associated disordered spin model. We derive the Replica Symmetry
Broken equations for a general class of disordered spin models which includes
the Sherrington - Kirkpatrick model, the Ising -spin model as well as the
overconstrained 3-SAT model as particular cases. We have numerically solved the
Replica Symmetry Broken equations using a pseudo-spectral code down to
and including zero temperature. We find that the UNSAT-phase of the
overconstrained 3-SAT model is of the -RSB kind: in order to get a
stable solution the replica symmetry has to be broken in a continuous way,
similarly to the SK model in external magnetic field.Comment: 19 pages, 7 figures; some section improved; iopart styl
Experimental realization of a highly structured search algorithm
The highly structured search algorithm proposed by Hogg[Phys.Rev.Lett.
80,2473(1998)] is implemented experimentally for the 1-SAT problem in a single
search step by using nuclear magnetic resonance technique with two-qubit
sample. It is the first demonstration of the Hogg's algorithm, and can be
readily extended to solving 1-SAT problem for more qubits in one step if the
appropriate samples possessing more qubits are experimentally feasible.Comment: RevTex, 11 pages + 3 pages of figure
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