An anatomy of a quantum adiabatic algorithm that transcends the Turing computability

Hilbert’s tenth problem 1 asks for a single procedure/algorithm to systematically determine if any given Diophantine equation has some positive integer solution or not. This problem has now been proved to be Turing noncomputable, indirectly through an equivalence mapping to the noncomputable Turing halting problem . Nevertheless, we have proposed and argued (with more and more details provided) in a series of papers 3,4,5,6,7 for an algorithm for Hilbert’s tenth problem based on quantum adiabatic computation . Among the many valid concerns about our algorithm there are also some misleading objections which are still being spread despite their falsehood. In this short note we will give an updated overall picture of the algorithm and go through, with pointers provided for further discussions elsewhere, some of the valid concerns as well as the misleading objections to dispel the entrenched but baseless prejudice against our proposed algorithm.