We review the proposal of a quantum algorithm for Hilbert's tenth problem and provide further arguments towards the proof that: (i) the algorithm terminates after a finite time for any input of Diophantine equation; (ii) the final ground state which contains the answer for the Diophantine equation can be identified as the component state having better-than-even probability to be found by measurement at the end time--even though probability for the final ground state in a quantum adiabatic process need not monotonically increase towards one in general. Presented finally are the reasons why our algorithm is outside the jurisdiction of no-go arguments previously employed to show that Hilbert's tenth problem is recursively non-computable.
To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically non-decidable problems, those of a modified version of the Hilbert's tenth problem and of the computation of the Chaitin's
Ω number, which is a representation of the Gödel's Incompletness theorem. Some interesting connection to Quantum Field Theory is pointed out.