Quantum Algorithm for Hilbert's Tenth Problem

@article{Kieu2001QuantumAF,
  title={Quantum Algorithm for Hilbert's Tenth Problem},
  author={Tien D. Kieu},
  journal={International Journal of Theoretical Physics},
  year={2001},
  volume={42},
  pages={1461-1478}
}
  • T. Kieu
  • Published 23 October 2001
  • Physics
  • International Journal of Theoretical Physics
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… 

Computing the non-computable

  • T. Kieu
  • Mathematics, Computer Science
    ArXiv
  • 2002
It is argued that computability, and thus the limits of mathematics, ought to be determined not solely by mathematics itself but also by physical principles.

Hypercomputation with quantum adiabatic processes

  • T. Kieu
  • Mathematics
    Theor. Comput. Sci.
  • 2004

Quantum Principles and Mathematical Computability

Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique

The quantum algorithm of Kieu does not solve the Hilbert's tenth problem

Recently T. Kieu (arXiv:quant-ph/0110136) claimed a quantum algorithm computing some functions beyond the Church-Turing class. He notes that "it is in fact widely believed that quantum computation

Hilbert ’ s Incompleteness , Chaitin ’ s Ω number and Quantum Physics

To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert’s tenth problem, we consider two further classes of mathematically nondecidable problems, those of a

A possible hypercomputational quantum algorithm

A possible quantum algorithm for a classically non-computable decision problem, Hilbert's tenth problem, is presented and a possible hypercomputation model based on quantum computation is presented.

Hypercomputation based on quantum computing

A hypercomputation model based on quantum computation based on the quantum adiabatic process and the characteristics of the representation of the dynamical algebra su(1,1) associated to the infinite square well is presented.

Construction of a universal quantum computer

A universal quantum computer that can emulate any classical Turing machine and can execute any algorithm that can be implemented in the quantum gate array framework but under the control of a quantum program, and hence is quantum computationally universal.

Computational Power of Infinite Quantum Parallelism

A new attempt towards hypercomputation is compared by separating its principal computing capabilities from realizability issues, which are shown to coincide with recursive enumerability and the second are considered in analogy to ‘existence’ in mathematical logic.

Does Quantum Mechanics allow for Infinite Parallelism?

A new attempt towards hypercomputation is compared by separating 1) its principal computing capabilities from 2) realizability issues, and it is shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic.
...

References

SHOWING 1-10 OF 38 REFERENCES

Computing the non-computable

  • T. Kieu
  • Mathematics, Computer Science
    ArXiv
  • 2002
It is argued that computability, and thus the limits of mathematics, ought to be determined not solely by mathematics itself but also by physical principles.

Numerical simulations of a quantum algorithm for Hilbert's tenth problem

  • T. Kieu
  • Mathematics
    SPIE Defense + Commercial Sensing
  • 2003
The Quantum Adiabatic Theorem enables us to establish a connection between the solution for this class of problems and the asymptotic behavior of solutions of a particular type of time-dependent Schrodinger equations.

Quantum complexity theory

This paper gives the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis, and proves that bits of precision suffice to support a step computation.

The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines

In this paper a microscopic quantum mechanical model of computers as represented by Turing machines is constructed. It is shown that for each numberN and Turing machineQ there exists a HamiltonianHNQ

Quantum algorithmic information theory

The agenda of quantum algorithmic information theory, ordered ‘top-down,’ is the quantum halting amplitude, followed by the quantumgorithmic information content, which in turn requires the theory of quantum computation, based upon a model of universal quantum computer whose elementary unit is a two-port interferometer capable of arbitrary U(2) transformations.

Coins, Quantum Measurements, and Turing's Barrier

A mathematical quantum “device” (with sensitivity ε) is constructed to solve the Halting Problem and proves the Wiener measure of Fε, Tconstructively tends to zero when T tends to infinity.

Measurability and Computability

The conceptual relation between the measurability of quantum mechanical observables and the computability of numerical functions is re-examined. A new formulation is given for the notion of

Comments on Adiabatic Quantum Algorithms

Recently a method for adiabatic quantum computation has been proposed and there has been considerable speculation about its efficiency for NP-complete problems. Heuristic arguments in its favor are

Quantum Computation by Adiabatic Evolution

We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that

Quantum Computation over Continuous Variables

This paper provides necessary and sufficient conditions for constructing a universal quantum computer over continuous variables. As an example, it is shown how a universal quantum computer for the