Feynman and computation: exploring the limits of computers
Feynman and computation: exploring the limits of computers
Reflections on quantum computing
Complexity
Computing with cells and atoms: an introduction to quantum, DNA and membrane computing
Computing with cells and atoms: an introduction to quantum, DNA and membrane computing
Ultimate zero and one: computing at the quantum frontier
Ultimate zero and one: computing at the quantum frontier
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
The Quantum Domain As a Triadic Relay
UMC '00 Proceedings of the Second International Conference on Unconventional Models of Computation
Incompleteness, Complexity, Randomness and Beyond
Minds and Machines
Hypercomputation: philosophical issues
Theoretical Computer Science - Super-recursive algorithms and hypercomputation
Zeno machines and hypercomputation
Theoretical Computer Science
A broader view on the limitations of information processing and communication by nature
Natural Computing: an international journal
Quantum computing: beyond the limits of conventional computation
International Journal of Parallel, Emergent and Distributed Systems - Emergent Computation
Hierarchy and equivalence of multi-letter quantum finite automata
Theoretical Computer Science
Finite automata models of quantized systems: conceptual status and outlook
DLT'02 Proceedings of the 6th international conference on Developments in language theory
Quantum query algorithms for conjunctions
UC'10 Proceedings of the 9th international conference on Unconventional computation
Prefix-Like complexities and computability in the limit
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Algorithmic randomness, quantum physics, and incompleteness
MCU'04 Proceedings of the 4th international conference on Machines, Computations, and Universality
| Hi-index | 0.00 |
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” 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.PACS: 03.67.Lx