Quantum Hypercomputation TIEN D. KIEU Centre for Atom Optics and Ultrafast Spectroscopy, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia; E-mail: kieu@swin.edu.au Abstract. We explore the possibility of using quantum mechanical principles for hypercomputation through the consideration of a quantum algorithm for computing the Turing halting problem. The mathematical noncomputability is compensated by the measurability of the values of quantum ob- servables and of the probability distributions for these values. Some previous no-go claims against quantum hypercomputation are then reviewed in the light of this new positive proposal. Key words: Hilbert’s tenth problem, quantum adiabatic theorem, quantum computation When we resolve a paradox, we do not decide in favor of one of the conflicting arguments and against the other; rather, we bring out the precise truth of each in order to show they do not conflict on the same ground. Michael Scriven (1964) ...we would be profoundly surprised if the physics of the real world can be properly and fully set out without departing from the set of Turing-machine- computable functions... In short it would – or should – be one of the greatest astonishments of science if the activity of Mother Nature were never to stray beyond the bounds of Turing-machine-computability. B.J. Copeland and R. Sylvan (1999) 1. Introduction Ever since the inception of the universal Turing computer – simple but yet encom- passingly powerful – there have been continuing efforts to understand its power and its limitations and to extend computation beyond such limitations. Supported by the convergence of many seemingly different models of compu- tation put forward independently by different people – Turing, Post, Markov and others (Lewis and Papadimitriou, 1981) – a thesis regarding the limits of comput- ability has been framed and has gained much credibility. The Church-Turing thesis can be phrased as Every function which would naturally be regarded as computable can be com- puted by a universal Turing machine. Minds and Machines 12: 541–561, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.