## 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

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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... |

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Citation Context ...o compute the noncomputables, provided certain hamiltonian and its ground state can be physically constructed. We propose a quantum algorithm for the classically noncomputable Hilbert’s tenth problem =-=[6]-=- which ∗ kieu@swin.edu.au 1ultimately links to the halting problem for Turing machines in the computation of partial recursive functions. The practical details of implementation the quantum algorithm... |

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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 ...f Quantum Mechanics. This also reconciles with the Cantor diagonal arguments which state that the problem could not be solved entirely in the framework of classical computation. Below is an algorithm =-=[17]-=- based on this philosophy of exploiting the interplay between the presumably infinite physical world and the theory of Quantum Mechanics calculated 8in a finite manner on Turing machines. The algorit... |

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Citation Context ...simulate the required hamiltonian and to obtain the ground state adiabatically. 6 Simulating the hamiltonians One way to construct any suitable hamiltonian so desired is through the technique of ref. =-=[14]-=-. We consider the hermitean operators, where j is the index of the unknowns of 6the Diophantine equation, Xj = 1 √ 2 (aj + a † j), Pj = i √ 2 (aj − a † j), (7) [Pj, Xk] = iδjk. Together with the avai... |

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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 ...in proof The author has recently obtained a criterion based on the final propbability distribution, which can be measured to arbitrary precision, for identifying the ground state of HP at time T. See =-=[28]-=- for more details. 13Acknowledgements I am indebted to Alan Head for discussions, comments and suggestions. I would also like to thank Cristian Calude, John Markham, Boris Pavlov, Andrew Rawlinson an... |

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Citation Context ...Russell resolved the set theory paradox (to do with “The set of all sets which are not members of themselves”) by the introduction of classes as distinct from sets. (For other lines of arguments, see =-=[9]-=-.) 4 An observation It suffices to consider nonnegative solutions, if any, of a Diophantine equation. Let us consider the example (x + 1) 3 + (y + 1) 3 − (z + 1) 3 + cxyz = 0, c ∈ Z, (3) with unknowns... |

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Citation Context ...or certain unitary processes cannot somehow be admitted as quantum dynamics. And up to now we do not have any evidence nor any principles that prohibit these kinds of observables and dynamics. (Ozawa =-=[13]-=- has produced some counter arguments but we think they are not quite applicable here. See [9].) Our general algorithm above could be realised by, but in no way restricted to, the following methods to ... |

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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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(Show Context)
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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- 2002
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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... |