## A reformulation of Hilbert’s tenth problem through Quantum Mechanics (2001)

Citations: | 16 - 9 self |

### BibTeX

@MISC{Kieu01areformulation,

author = {Tien D Kieu},

title = {A reformulation of Hilbert’s tenth problem through Quantum Mechanics},

year = {2001}

}

### OpenURL

### Abstract

Inspired by Quantum Mechanics, we reformulate Hilbert’s tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shrödinger propagator with some appropriate kernel. Either way, Mathematics and Physics could be combined for Hilbert’s tenth problem and for the notion of effective computability. 1

### Citations

837 |
Theory of recursive functions and effective computability
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- 1967
(Show Context)
Citation Context ...lished within itself, through the noncomputable/undecidable results of Hilbert’s tenth problem, Gödel’s incompleteness theorem, Turing halting problem, and their various extensions (see, for example, =-=[1, 2]-=-). Such noncomputability and undecidability set the boundary for computation carried out by mechanical processes, and in doing so it help us to understand much better what can be so computed mathemati... |

562 |
A Decision Method for Elementary Algebra and Geometry, University of California Press
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(Show Context)
Citation Context ...t seems –because the unsolvability of Hilbert’s tenth problem is only established in the framework of integer arithmetic and in Turing computability, not necessarily in Mathematics in general. Tarski =-=[7]-=- has shown that the question about the existence of real solutions of polynomials over the reals is, in fact, decidable. In the case of the linear Schrödinger (30), we could also exploit a powerful co... |

479 | Quantum complexity theory
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- 1997
(Show Context)
Citation Context ...le out the crossing of other states different from the ground state.) It should also be noted that the above quantum mechanical approach to Turingnoncomputable problems is in contrast to the claim in =-=[11]-=- that quantum Turing machines compute exactly the same class of functions, albeit perhaps more efficiently, which can be computed by classical Turing machines. However, the quantum Turing machine appr... |

393 | Simulating physics with computers
- Feynman
(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 =-=[12, 13]-=- in the conception days of quantum computation. Indeed, Nielsen [14] has also found no logical contradiction in applying the most general quantum mechanical principles to the computation of the classi... |

146 |
Hilbert’s tenth problem
- Matiyasevich
- 1993
(Show Context)
Citation Context ...d in doing so it help us to understand much better what can be so computed mathematically. We have proposed elsewhere [3] a quantum algorithmic approach for the non-computable Hilbert’s tenth problem =-=[1, 4]-=-, which is equivalent to the Turing halting problem and intimately links to the concept of effective computability as defined by the Church-Turing thesis. While the proposal is about some quantum proc... |

114 |
Computability and Unsolvability
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(Show Context)
Citation Context ...lished within itself, through the noncomputable/undecidable results of Hilbert’s tenth problem, Gödel’s incompleteness theorem, Turing halting problem, and their various extensions (see, for example, =-=[1, 2]-=-). Such noncomputability and undecidability set the boundary for computation carried out by mechanical processes, and in doing so it help us to understand much better what can be so computed mathemati... |

60 | Quantum algorithm for hilbert’s tenth problem
- Kieu
(Show Context)
Citation Context ...idability set the boundary for computation carried out by mechanical processes, and in doing so it help us to understand much better what can be so computed mathematically. We have proposed elsewhere =-=[3]-=- a quantum algorithmic approach for the non-computable Hilbert’s tenth problem [1, 4], which is equivalent to the Turing halting problem and intimately links to the concept of effective computability ... |

51 | Quantum computation by adiabatic evolution
- Farhi, Goldstone, et al.
- 2000
(Show Context)
Citation Context ...ason. It should be a symmetry reason, that is, H(s0) should commute with some other hermitean operator(s). Our mathematical elaboration above agrees with the observation of symmetry and degeneracy in =-=[8]-=-. With care we can slightly modify the derivation for (13, 14) to come up with similar equations even when there is some degeneracy in [0, 1]. But for the condition (15) to be the indicator for the ex... |

11 |
Quantum Computation by Adiabatic Evolution, arXiv: quant-ph/0001106v1
- Farhi, Goldstone, et al.
(Show Context)
Citation Context ...ason. It should be a symmetry reason, that is, H(s0) should commute with some other hermitean operator(s). Our mathematical elaboration above agrees with the observation of symmetry and degeneracy in =-=[6]-=-. With care we can slightly modify the derivation for (11,12) to come up with similar equations even when there is some degeneracy in [0, 1]. But for the condition (13) to be the indicator for the exi... |

7 | On the existence of a new family of Diophantine equations for Ω
- Ord, Kieu
(Show Context)
Citation Context ..., which is at the foundation of computability. The undecidability result is thus singularly important: Hilbert’s tenth problem could be solved if and only if could be the Turing halting problem. (See =-=[5]-=- also.) Given a Diophantine equation with K unknowns x’s D(x1, · · ·, xK) = 0, (1) it suffices in general to consider the existence of nonnegative integer solutions. Following [3] we link the equation... |

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 =-=[12, 13]-=- in the conception days of quantum computation. Indeed, Nielsen [14] has also found no logical contradiction in applying the most general quantum mechanical principles to the computation of the classi... |

3 |
Quantum Mechanics, 2nd Edition
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- 1976
(Show Context)
Citation Context ...alues, say (l, l+1) out of the infinitely many, can come very close together then they can never actually cross. The arguments are similar to those of perturbation theory for nearly degenerate levels =-=[6]-=-. As can be seen from (13), in a neighbourhood around s0 we need only consider the two states |El〉 and |El+1〉 as they are so strongly coupled that the rest can be safely ignored. As our world just bec... |

2 | Comments on adiabatic quantum algorithms
- Ruskai
- 2002
(Show Context)
Citation Context ...quire no level crossing for the instantaneous ground state, unlike the situation with the nonlinear equations previously where we have required no crossing for all levels. Adapting Ruskai’s arguments =-=[10]-=- we can show that the ground state of (3) is non-degenerate for s ∈ (0, 1). (The arguments are only applicable for the ground state but, interestingly, the conclusion of the last section can also be s... |

2 | Hilbert’s incompleteness, Chaitin’s Ω number and quantum physics, Los Alamos preprint archive http://arXiv:quant-ph/0111062
- Kieu
- 2001
(Show Context)
Citation Context ...added as extra ingredients. In particular, approximate numerical results would be needed in an estimation of the adiabatic time for a physical implementation of the quantum algorithm in [1] (see also =-=[7]-=-) for the Hilbert’s tenth problem. In doing so, we have to assume that Nature is describable by Quantum Mechanics correctly at least to the precision required. If not, testing a known Diophantine equa... |