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Hypercomputability of Quantum Adiabatic Processes
Tien D. Kieu

We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented.We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementations. We also point out a surprising connection between quantum computability and quantum phase transitions through some instances of our quantum adiabatic algorithm.

Keywords: Computability, Hypercomputability, Hilbert’s Tenth Problem, Diophantine Equations, Quantum Computation, Quantum Adiabatic Computation, Quantum Phase Transitions.

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