## Computing a Turing-incomputable problem from quantum computing

Venue: | quant-ph/0309198 (2003). 12, 2008 9:5 WSPC/INSTRUCTION FILE Transcending2 Transcending Turing computability 7 |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Sicard_computinga,

author = {Andrés Sicard and Mario Vélez and Juan Ospina},

title = {Computing a Turing-incomputable problem from quantum computing},

booktitle = {quant-ph/0309198 (2003). 12, 2008 9:5 WSPC/INSTRUCTION FILE Transcending2 Transcending Turing computability 7},

year = {}

}

### OpenURL

### Abstract

Abstract. A hypercomputation model named Infinite Square Well Hypercomputation Model (ISWHM) is built from quantum computation. This model is inspired by the model proposed by Tien D. Kieu [1] and solves an Turing-incomputable problem. For the proposed model and problem, a simulation of its behavior is made. Furthermore, it is demonstrated that ISWHM is a universal quantum computation model. 1

### Citations

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Citation Context ...tine equations has a solution in ZZ, if and only if a certain Diophantine equation associated to the system has a solution in ZZ; the Hilbert’s tenth problem may be reduced to search a solution in IN =-=[6]-=-. Due to the Hamiltonian operator (2) used in the description of ISWHM, the decision problem associated with this model of computation consists in determining whether or not a Diophantine equation kin... |

35 | Hypercomputation: computing more than the Turing machine, Honours Thesis
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Citation Context ...e M thesis, which identifies the functions calculable by a machine with the Turingcomputable functions [4]. On the other hand, notwithstanding the proliferation of theoretical hypercomputation models =-=[5]-=-, the possibility of the real construction of a hypermachine keeps being controversial and under analysis. 3 The Turing-Incomputable Problem The demonstration that ISWHM is a hypercomputation model wi... |

28 | On universal fault-tolerant quantum computing: A novel basis and a new constructive proof of universality for shor’s basis
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Citation Context ...e defined. To demonstrate the universality of the ISWHM it is necessary and sufficient , the phase to construct three quantum gates: the controlled-NOT gate C4 NOT gate Φ2 π/4 and Hadamard gate H2 ad =-=[9,10]-=-. For gates C4 NOT and Φ2 π/4 , its matrix representation and the transformation carried out upon the base states of Hilbert space on which they act, are given by C 4 NOT = ⎛ ⎞ 1 0 0 0 ⎜0 1 0 0 ⎟ ⎝0 0... |

16 | Computation and hypercomputation
- Stannett
- 2003
(Show Context)
Citation Context ...at cannot be performed by a Turing machine [2]. A hypermachine, also capable of simulating a universal Turing machine is called a super-(Turing machine), otherwise it is called a non-(Turing machine) =-=[3]-=-. At first glance, we could think that the possibility of a hypercomputation model would be a refutation of the widely accepted Church-Turing thesis, which identifies the naturally calculable function... |

9 | Numerical simulations of a quantum algorithm for Hilbert’s tenth problem, available in electronic form from http://arxiv.org/PS cache/quant-ph/pdf/0304/0304114.pdf
- Kieu
(Show Context)
Citation Context ...HA(t/T) = (1 − t/T)HI + (t/T)HD , where T is the total time of the adiabatic evolution and t ∈ [0, T]. The Hamiltonian HA(t/T) determines the solution of the Schrödinger discretized equation given by =-=[8]-=- ψ(t + δt) = ( 1 − i 2 HA(t/T)δt 1 + i 2 HA(t/T)δt ) ψ(t) , (4) where δt is the time step. For a Diophantine equation D(x) = 0, based on (4) the algorithm for ISWHM to determine whether or not it has ... |

5 |
Hipercomputación: la próxima generación de la computación teórica (Hypercomputation: the next generation of theoretical computation
- Sicard, Vélez
(Show Context)
Citation Context ... the Turing-computable functions. But actually the existence of hypercomputation models refute the M thesis, which identifies the functions calculable by a machine with the Turingcomputable functions =-=[4]-=-. On the other hand, notwithstanding the proliferation of theoretical hypercomputation models [5], the possibility of the real construction of a hypermachine keeps being controversial and under analys... |

4 |
Mecánica Cuántica
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(Show Context)
Citation Context ...the decision problem associated with ISWHM is a Turing-incomputable one.4 The Physical Referent The energy levels En, for a particle with mass m in a infinite square well with length L, are given by =-=[7]-=- En = ¯h2 π 2 2mL 2n2 . These energy levels are the eigenvalues of the Hamiltonian operator of the system denoted by H, for which correspond as eigenvectors the states denoted by the kets | n〉, where ... |

1 |
Computing the noncomputable. Contemporay Physics 44 (2003) 51–71 Eprint: arXiv.org/abs/quant-ph/0203034
- Kieu
- 1999
(Show Context)
Citation Context ...epm.net.co Abstract. A hypercomputation model named Infinite Square Well Hypercomputation Model (ISWHM) is built from quantum computation. This model is inspired by the model proposed by Tien D. Kieu =-=[1]-=- and solves an Turing-incomputable problem. For the proposed model and problem, a simulation of its behavior is made. Furthermore, it is demonstrated that ISWHM is a universal quantum computation mode... |

1 |
M.: Modelos de computación cuántica discreta. En: Informe de investigación [11]. Preprint: sigma.eafit.edu.co:90/~asicard/archivos/colombia.ps.gz
- Vélez, Sicard, et al.
- 2000
(Show Context)
Citation Context ...e defined. To demonstrate the universality of the ISWHM it is necessary and sufficient , the phase to construct three quantum gates: the controlled-NOT gate C4 NOT gate Φ2 π/4 and Hadamard gate H2 ad =-=[9,10]-=-. For gates C4 NOT and Φ2 π/4 , its matrix representation and the transformation carried out upon the base states of Hilbert space on which they act, are given by C 4 NOT = ⎛ ⎞ 1 0 0 0 ⎜0 1 0 0 ⎟ ⎝0 0... |

1 |
M.: Prototipo de un modelo de computación cuántica continua
- Sicard, Vélez
(Show Context)
Citation Context ...volution unitary operator and | ψ(0)〉 = | n〉 ∞ is considered for the eigenstates of the infinite square well Hamiltonian. The U(t) matrix elements are given by ( Unp(t) = exp −i ¯hπ2 n2t 2mL2 ) δnp , =-=(11)-=- where δnp is Kronecker delta. From different choices of the t parameter in (11), and different qubits coding, it is possible to find the evolutions corresponding to each of the mentioned quantum gate... |