2,755 research outputs found
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
A quantum system will stay near its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough. This quantum
adiabatic behavior is the basis of a new class of algorithms for quantum
computing. We test one such algorithm by applying it to randomly generated,
hard, instances of an NP-complete problem. For the small examples that we can
simulate, the quantum adiabatic algorithm works well, and provides evidence
that quantum computers (if large ones can be built) may be able to outperform
ordinary computers on hard sets of instances of NP-complete problems.Comment: 15 pages, 6 figures, email correspondence to [email protected] ; a
shorter version of this article appeared in the April 20, 2001 issue of
Science; see http://www.sciencemag.org/cgi/content/full/292/5516/47
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
Ising formulations of many NP problems
We provide Ising formulations for many NP-complete and NP-hard problems,
including all of Karp's 21 NP-complete problems. This collects and extends
mappings to the Ising model from partitioning, covering and satisfiability. In
each case, the required number of spins is at most cubic in the size of the
problem. This work may be useful in designing adiabatic quantum optimization
algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more
references; v3: substantial revision and extension, to-be-published versio
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