dorsal/arxiv
View SchemaCoins, Quantum Measurements, and Turing's Barrier
| Authors | Cristian S. Calude, Boris Pavlov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0112087 |
| URL | https://arxiv.org/abs/quant-ph/0112087 |
| Journal | Quantum Information Processing, 1, 1--2 (2002), 107--127 |
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.}
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"abstract": "Is there any hope for quantum computing to challenge the Turing barrier, i.e.\nto solve an undecidable problem, to compute an uncomputable function? According\nto Feynman\u0027s \u002782 argument, the answer is {\\it negative}. This paper re-opens\nthe case: we will discuss solutions to a few simple problems which suggest that\n{\\it quantum computing is {\\it theoretically} capable of computing uncomputable\nfunctions}. In this paper a mathematical quantum \"device\" (with sensitivity\n$\\epsilon$) is constructed to solve the Halting Problem. The \"device\" works on\na randomly chosen test-vector for $T$ units of time. If the \"device\" produces a\nclick, then the program halts. If it does not produce a click, then either the\nprogram does not halt or the test-vector has been chosen from an {\\it\nundistinguishable set of vectors} ${\\IF}_{\\epsilon, T}$. The last case is not\ndangerous as our main result proves: {\\it the Wiener measure of}\n${\\IF}_{\\epsilon, T}$ {\\it constructively tends to zero when} $T$ {\\it tends to\ninfinity}. The \"device\", working in time $T$, appropriately computed, will\ndetermine with a pre-established precision whether an arbitrary program halts\nor not. {\\it Building the \"halting machine\" is mathematically possible.}",
"arxiv_id": "quant-ph/0112087",
"authors": [
"Cristian S. Calude",
"Boris Pavlov"
],
"categories": [
"quant-ph"
],
"journal_ref": "Quantum Information Processing, 1, 1--2 (2002), 107--127",
"title": "Coins, Quantum Measurements, and Turing\u0027s Barrier",
"url": "https://arxiv.org/abs/quant-ph/0112087"
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