26,936 research outputs found
NP-complete Problems and Physical Reality
Can NP-complete problems be solved efficiently in the physical universe? I
survey proposals including soap bubbles, protein folding, quantum computing,
quantum advice, quantum adiabatic algorithms, quantum-mechanical
nonlinearities, hidden variables, relativistic time dilation, analog computing,
Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and
"anthropic computing." The section on soap bubbles even includes some
"experimental" results. While I do not believe that any of the proposals will
let us solve NP-complete problems efficiently, I argue that by studying them,
we can learn something not only about computation but also about physics.Comment: 23 pages, minor correction
Quantum Portfolios
Quantum computation holds promise for the solution of many intractable
problems. However, since many quantum algorithms are stochastic in nature they
can only find the solution of hard problems probabilistically. Thus the
efficiency of the algorithms has to be characterized both by the expected time
to completion {\it and} the associated variance. In order to minimize both the
running time and its uncertainty, we show that portfolios of quantum algorithms
analogous to those of finance can outperform single algorithms when applied to
the NP-complete problems such as 3-SAT.Comment: revision includes additional data and corrects minor typo
Quantum Computational Complexity in the Presence of Closed Timelike Curves
Quantum computation with quantum data that can traverse closed timelike
curves represents a new physical model of computation. We argue that a model of
quantum computation in the presence of closed timelike curves can be formulated
which represents a valid quantification of resources given the ability to
construct compact regions of closed timelike curves. The notion of
self-consistent evolution for quantum computers whose components follow closed
timelike curves, as pointed out by Deutsch [Phys. Rev. D {\bf 44}, 3197
(1991)], implies that the evolution of the chronology respecting components
which interact with the closed timelike curve components is nonlinear. We
demonstrate that this nonlinearity can be used to efficiently solve
computational problems which are generally thought to be intractable. In
particular we demonstrate that a quantum computer which has access to closed
timelike curve qubits can solve NP-complete problems with only a polynomial
number of quantum gates.Comment: 8 pages, 2 figures. Minor changes and typos fixed. Reference adde
Quantum Algorithm for SAT Problem and Quantum Mutual Entropy
It is von Neumann who opened the window for today's Information epoch. He
defined quantum entropy including Shannon's information more than 20 years
ahead of Shannon, and he introduced a concept what computation means
mathematically. In this paper I will report two works that we have recently
done, one of which is on quantum algorithum in generalized sense solving the
SAT problem (one of NP complete problems) and another is on quantum mutual
entropy properly describing quantum communication processes.Comment: 19 pages, Proceedings of the von Neumann Centennial Conference:
Linear Operators and Foundations of Quantum Mechanics, Budapest, Hungary,
15-20 October, 200
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