Can quantum computing solve classically unsolvable problems?

T. D. Kieu has claimed that a quantum computing procedure can solve a classically unsolvable problem. Recent work of W. D. Smith has shown that Kieu’s central mathematical claim cannot be sustained. Here, a more general critique is given of Kieu’s proposal and some suggestions are made regarding the Church-Turing thesis. 0. Introduction: In 2001 the physicist Tien D. Kieu announced that a quantum computing procedure could solve a classically unsolvable problem, namely Hilbert’s Tenth Problem. This problem, that of deciding whether a polynomial integer-valued function of integers ever vanishes, is essentially equivalent to the standard ‘halting problem’ and other such unsolvable problems as defined by Turing machines. (The classic exposition of this fascinating connection is by Martin Davis (1958, 1982)). Kieu has since produced a number of publications with overlapping material. Reference will be made in what follows to some of the arXiv versions of his exposition. But it should be noted that his work has also been accepted and published in leading journals. Kieu’s claim is closely connected with the recent programme of so-called ‘hypercomputation’ as laid out by Copeland, Stannett, Ord, Calude and others (for a recent review see (Stannett 2004)). According to this view, as expounded in Scientific American by the philosophers Copeland and Proudfoot (1999), the computing of classically uncomputable functions (‘hypercomputing’) may be a matter of finding the right technology to make it possible. Indeed these authors enthusiastically announced the search for such technology to be already under way. Kieu’s proposal emerged as perhaps the brightest hope for fulfilling this dream. It is also the strongest claim made for power of quantum computation. These