57 research outputs found
An anatomy of a quantum adiabatic algorithm that transcends the Turing computability
We give an update on a quantum adiabatic algorithm for the Turing
noncomputable Hilbert's tenth problem, and briefly go over some relevant issues
and misleading objections to the algorithm.Comment: 7 pages, no figure. Submitted to the Proceedings of the conference
"Foundations of Quantum Information" (April 2004, Camerino, Italy
A reformulation of Hilbert's tenth problem through Quantum Mechanics
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the
domain of integer arithmetics into either a problem involving a set of
infinitely coupled differential equations or a problem involving a Shr\"odinger
propagator with some appropriate kernel. Either way, Mathematics and Physics
could be combined for Hilbert's tenth problem and for the notion of effective
computability
Numerical simulations of a quantum algorithm for Hilbert's tenth problem
We employ quantum mechanical principles in the computability exploration of
the class of classically noncomputable Hilbert's tenth problem which is
equivalent to the Turing halting problem in Computer Science. The Quantum
Adiabatic Theorem enables us to establish a connection between the solution for
this class of problems and the asymptotic behaviour of solutions of a
particular type of time-dependent Schr\"odinger equations. We then present some
preliminary numerical simulation results for the quantum adiabatic processes
corresponding to various Diophantine equations.Comment: Oral contribution to SPIE, Orlando, 2003. 7 pages, 6 colour figure
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