Quantum adiabatic algorithm for Hilbert's tenth problem: I. The algorithm

TL;DR

This paper reviews a quantum adiabatic algorithm designed to solve Hilbert's tenth problem, providing arguments for its finite termination and the identification of the solution state through measurement, challenging previous non-computability claims.

Contribution

The paper introduces a quantum adiabatic algorithm for Hilbert's tenth problem and argues for its finite termination and solution identification, countering no-go theorems.

Findings

Algorithm terminates after finite time for any Diophantine input

Final ground state can be identified via measurement with better-than-even probability

Provides reasons why the algorithm evades previous non-computability proofs

Abstract

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.