## 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 ...class of classically noncomputable functions, such as the halting problem for Turing machines [4]. It is in fact widely believed that quantum computation cannot offer anything new about computability =-=[5]-=-. Contrary to this, we propose that quantum computation may be able to compute the noncomputables, provided certain hamiltonian and its ground state can be physically constructed. We propose a quantum... |

428 | Simulating physics with computers
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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 ...ers. It is clear even in the case of a single qubit that the state α|0〉+β|1〉 cannot be encoded as integers for all α and β - simply because of different cardinalities. In fact, the no-cloning theorem =-=[8]-=- of quantum mechanics does restrict the type of operations available to quantum algorithms. In essence, the way we will break the self-referential reasoning here by the differentiation between quantum... |

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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 ...equation has integer solutions. This decision problem for such polynomial equations, which are also known as Diophantine equations, has eventually been shown in 1970 by Matiyasevich to be undecidable =-=[6, 7]-=- in the Turing sense. It is consequently noncomputable/undecidable in the most general sense if one accepts, as almost everyone does, the Church-Turing thesis of computability. Since exponential Dioph... |

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Citation Context ...xample 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 computation could help solving the previous undec... |

54 | 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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34 |
Probability Theory (North-Holland
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Citation Context ...ability distribution P(T; ǫ) at the time T with an accuracy ǫ for all the measured states. The convergence of this repetition process is ensured by the Weak Law of Large Numbers in probability theory =-=[18]-=-. (An overestimate of the number of repetitions is L ≥ 1/(ǫ 2 (1 − p)).) Note the lowest energy state so obtained, |⃗nc〉, as the candidate ground state. • Step 3 (on the classical computer): Choose a ... |

31 | Computing the noncomputable
- Kieu
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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... |

23 |
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(Show Context)
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... |

22 |
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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... |

16 | A reformulation of Hilbert’s tenth problem through quantum mechanics
- Kieu
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(Show Context)
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... |

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
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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... |

8 |
The diagonal method and hypercomputation
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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... |

6 | Measurability and computability
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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 ... |

6 | Quantum annealing in the transverse Ising model
- Kadowaki, Nishimori
- 1998
(Show Context)
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
- Calude, Pavlov
- 2002
(Show Context)
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
(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
- Braunstein
- 1998
(Show Context)
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
- Ruskai
- 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... |