Coins, Quantum Measurements, and Turing's Barrier

Is there any hope for quantum computing to challenge the Turing barrier, i.e. to solve an undecidable problem, to compute an uncomputable function? According to Feynman’s ’82 argument, the answer is negative. This paper re-opens the case: we will discuss solutions to a few simple problems which suggest that quantum computing is theoretically capable of computing uncomputable functions. Turing proved that there is no “halting (Turing) machine” capable of distinguishing between halting and non-halting programs (undecidability of the Halting Problem). Halting programs can be recognized by simply running them; the main difficulty is to detect non-halting programs. In this paper a mathematical quantum “device” (with sensitivity ε) is constructed to solve the Halting Problem. The “device” ∗A preliminary version of this paper has appeared in [9]. †Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, New Zealand. E-mail: [email protected]. ‡Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand. E-mail: [email protected]. works on a randomly chosen test-vector for T units of time. If the “device” produces a click, then the program halts. If it does not produce a click, then either the program does not halt or the test-vector has been chosen from an undistinguishable set of vectors Fε,T . The last case is not dangerous as our main result proves: the Wiener measure of Fε,T constructively tends to zero when T tends to infinity. The “device”, working in time T , appropriately computed, will determine with a pre-established precision whether an arbitrary program halts or not. Building the “halting machine” is mathematically possible. To construct our “device” we use the quadratic form of an iterated map (encoding the whole data in an infinite superposition) acting on randomly chosen vectors viewed as special trajectories of two Markov processes working in two different scales of time. The evolution is described by an unbounded, exponentially growing semigroup; finally a single measurement produces the result.