## Quantum Algorithm For Hilberts Tenth Problem (2003)

Venue: | Int.J.Theor.Phys |

Citations: | 60 - 10 self |

### BibTeX

@ARTICLE{Kieu03quantumalgorithm,

author = {Tien D Kieu},

title = {Quantum Algorithm For Hilberts Tenth Problem},

journal = {Int.J.Theor.Phys},

year = {2003},

pages = {1461--1478}

}

### Years of Citing Articles

### OpenURL

### Abstract

We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert’s tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the Church-Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles. 1

### Citations

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146 |
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52 | Quantum computation by adiabatic evolution
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Citation Context ... allows us to access the ground state will suffice. One way is perhaps to use that of quantum annealing or cooling [15]. Another way is to employ the quantum computation method of adiabatic evolution =-=[16]-=-. 7 Adiabatic quantum evolution In the adiabatic approach, one starts with a hamiltonian HI whose ground state |gI〉 is readily achievable. Then one forms the slowly varying hamiltonian H(s), s = t , w... |

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Citation Context ... correction protocol for continuous variables [22] which could be of help here to protect the wave functions from decoherence. However, the adiabatic computation we exploit is quite robust in general =-=[23]-=-. An imperfect conventional quantum algorithm might have different sorts of errors than an 11imperfect adiabatic process, where the system is kept close to the instantaneous ground state over time. D... |

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Citation Context ...l ground state. Our point is on computability and not on computational complexity, which depends on individual polynomials. Computability is based on the arguments that the adiabatic time T is finite =-=[19, 20]-=- (for a high probability of achieving the ground state) and that the ground state can be verified by employing the theory of Quantum Mechanics. As long as the energy gap is finite so is the computatio... |

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Citation Context ...nomial hamiltonian constructed, we need to obtain its ground state. Any approach that allows us to access the ground state will suffice. One way is perhaps to use that of quantum annealing or cooling =-=[15]-=-. Another way is to employ the quantum computation method of adiabatic evolution [16]. 7 Adiabatic quantum evolution In the adiabatic approach, one starts with a hamiltonian HI whose ground state |gI〉... |

5 |
quantum measurements, and Turing’s barrier
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Citation Context ...s and/or physical resources then it would be an example of information being limited by physics, rather than by logical arguments alone. Our study is an illustration of “Information is physical” (see =-=[24]-=- for another quantum mechanical approach and [25] for where the theory of General Relativity is also exploited for the computation of Turing noncomputables). That some generalisation of the notion of ... |

4 |
The computer as a physical system
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- 1980
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Citation Context ...nite unitary operators of the standard quantum Turing computation cannot do in a finite number of steps. Note that it was the general hamiltonian computation that was discussed by Benioff and Feynman =-=[10, 11]-=- in the conception days of quantum computation. Indeed, Nielsen [12] has also found no logical contradiction in applying the most general quantum mechanical principles to the computation of the classi... |

3 |
Error correction for continuous variables
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- 1998
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Citation Context ...rection Our approach above is in fact a combination of the quantum computation of continuous variables and of adiabatic evolution. There exists some error correction protocol for continuous variables =-=[22]-=- which could be of help here to protect the wave functions from decoherence. However, the adiabatic computation we exploit is quite robust in general [23]. An imperfect conventional quantum algorithm ... |

2 | Comments on adiabatic quantum algorithms
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Citation Context ...l ground state. Our point is on computability and not on computational complexity, which depends on individual polynomials. Computability is based on the arguments that the adiabatic time T is finite =-=[19, 20]-=- (for a high probability of achieving the ground state) and that the ground state can be verified by employing the theory of Quantum Mechanics. As long as the energy gap is finite so is the computatio... |

1 |
as privately communicated by Cristian Calude
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Citation Context ...em cannot be solved by classical computation, we will have to resort to quantum computation without a priori knowing exactly the time T, except the knowledge that it is finite. Constructive logicians =-=[21]-=- allow for this algorithmic situation under the so-called Markov’s Principle. 9 Discussion of the algorithm The quantum algorithm above can be proved to terminate (even though it could be after a very... |