  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.

Quantum Physics