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Coins, Quantum Measurements, and Turing's Barrier | | Cristian S. Calude
; Boris Pavlov
; | | Date: |
15 Dec 2001 | | Journal: | Quantum Information Processing, 1, 1--2 (2002), 107--127 | | Subject: | quant-ph | | 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 {it negative}. This paper re-opens the case: we will discuss solutions to a few simple problems which suggest that {it quantum computing is {it theoretically} capable of computing uncomputable functions}. In this paper a mathematical quantum "device" (with sensitivity $epsilon$) 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 {it undistinguishable set of vectors} ${IF}_{epsilon, T}$. The last case is not dangerous as our main result proves: {it the Wiener measure of} ${IF}_{epsilon, T}$ {it constructively tends to zero when} $T$ {it 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. {it Building the "halting machine" is mathematically possible.} | | Source: | arXiv, quant-ph/0112087 | | Services: | Forum | Review | PDF | Favorites | |
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