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  Computability of Hilbert's Tenth Problem via quantum adiabatic computation

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I recently stumbled upon a series of papers by Tien D. Kieu that claim that their proposed quantum algorithm solves Hilbert's Tenth Problem, that is, that the verification of the existence or nonexistence of solutions to arbitrary Diophantine equations is computable via quantum adiabatic computing.

This isn't my area of expertise, so I was hoping I could get the opinion of a more knowledgeable person about the quality of this work. The research seems to have generally gone unnoticed. Certain papers contest Kieu's findings, and Kieu has replied in turn.

The algorithm seems to be described here (and in various other papers as well).

asked Apr 4, 2015 in Mathematics by lumpenproletarian (15 points) [ revision history ]
retagged Apr 4, 2015

1 Answer

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A physical process can never solve a mathematical problem in Hilbert's sense, unless the process is 100% deterministic. Thus quantum algorithms have only a heuristic value unless they happen to produce a solution that can be verified independently.

A quantum algorithm produces answers with certain probabilities only. So instead of answering yes or no a quantum decision algorithm gives a probability that tends (with probability 1) to 0 or 1. At any fixed time, we have no certainty. In particular, unless the algorithm spits out a proof of nonexistence - which Kieu's algorithm doesn't - that can be checked by hand or with a deterministic proof checker, we'll never know whether a solution exist in case in fact none exists. 

The same applies for occasionally proposed quantum solutions of the still unsolved 'P=NP?' problem, one of the Clay Millennium problems.

answered Apr 4, 2015 by Arnold Neumaier (13,189 points) [ revision history ]

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