Abstract:
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 ") is constructed to solve the Halting Problem. The
“device” 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 F",T . The last case is not dangerous as our main
result proves: the Wiener measure of F",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.
