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

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

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