16,494 research outputs found
Solving Highly Constrained Search Problems with Quantum Computers
A previously developed quantum search algorithm for solving 1-SAT problems in
a single step is generalized to apply to a range of highly constrained k-SAT
problems. We identify a bound on the number of clauses in satisfiability
problems for which the generalized algorithm can find a solution in a constant
number of steps as the number of variables increases. This performance
contrasts with the linear growth in the number of steps required by the best
classical algorithms, and the exponential number required by classical and
quantum methods that ignore the problem structure. In some cases, the algorithm
can also guarantee that insoluble problems in fact have no solutions, unlike
previously proposed quantum search algorithms
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
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
A Framework for Structured Quantum Search
A quantum algorithm for general combinatorial search that uses the underlying
structure of the search space to increase the probability of finding a solution
is presented. This algorithm shows how coherent quantum systems can be matched
to the underlying structure of abstract search spaces, and is analytically
simpler than previous structured search methods. The algorithm is evaluated
empirically with a variety of search problems, and shown to be particularly
effective for searches with many constraints. Furthermore, the algorithm
provides a simple framework for utilizing search heuristics. It also exhibits
the same phase transition in search difficulty as found for sophisticated
classical search methods, indicating it is effectively using the problem
structure.Comment: 18 pages, Latex, 7 figures, further information available at
ftp://parcftp.xerox.com/pub/dynamics/quantum.htm
Experimental Implementation of Hogg's Algorithm on a Three-Quantum-bit NMR Quantum Computer
Using nuclear magnetic resonance (NMR) techniques with three-qubit sample, we
have experimentally implemented the highly structured algorithm for the 1-SAT
problem proposed by Hogg. A simplified temporal averaging procedure was
employed to the three-qubit spin pseudo-pure state. The algorithm was completed
with only a single evaluation of structure of the problem and the solutions
were found with probability 100%, which outperform both unstructured quantum
and the best classical search algorithm.Comment: Revtex, 14 pages and 1 table, 4 EPS figure
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