Computing the non-computable

Kieu, Tien D.

doi:10.1080/00107510302712

Cited by 42 publications

(31 citation statements)

References 14 publications

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“…When the parameter α increases adiabatically according to the adiabatic theorem (see, a detail discussion of the application of this theorem to quantum adiabatic computations given in Ref. [33]) the system will remain in the ground state. For small temperature the domain walls and domains may still proliferate into the system but if they do, their number is very small.…”

Section: Adiabatic Quantum Computations With a Planar Array Of π-mentioning

confidence: 99%

Two-dimensional Ising model with competing interactions and its application to clusters and arrays ofπ-rings and adiabatic quantum computing

et al. 2007

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We study planar clusters consisting of loops including a Josephson π-junction (π-rings). Each π-ring carries a persistent current and behaves as a classical orbital moment. The type of particular state associated with the orientation of orbital moments at the cluster depends on the interaction between these orbital moments and can be easily controlled, i.e. by a bias current or by other means. We show that these systems can be described by the two-dimensional Ising model with

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Section: Adiabatic Quantum Computations With a Planar Array Of π-mentioning

confidence: 99%

Two-dimensional Ising model with competing interactions and its application to clusters and arrays ofπ-rings and adiabatic quantum computing

et al. 2007

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“…The present work aims to put some reason into the ongoing, and often sensational [Kie03b,Lloy06], discussion. We are convinced that this requires formalizing Question 1.…”

Section: Discussionmentioning

confidence: 99%

Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

Ziegler

2009

Applied Mathematics and Computation

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Abstract. We turn 'the' Church-Turing Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and well-defined scientific problem(s):Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory-rather than vaguely referring to "nature" in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability and complexity in computational physics.

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“…Recently Tien D. Kieu has proposed an quantum algorithm to solve the TM incomputable ‡ problem named Hilbert's tenth problem, using as physical referent the well known simple harmonic oscillator (SHO), which by effect of the second quantization has as associated dynamical algebra the Weyl-Heisenberg algebra denoted g W−H [8,9,10,11,12,13]. From the algebraic analysis of Kieu's hypercomputational quantum algorithm (KHQA), we have identified the underlying properties of the g W−H algebra which are necessary (but not sufficient) to guaranty KHQA works.…”

Section: Introductionmentioning

confidence: 99%

“…Starting from that seems to be the first published model of hypercomputation, which is called the Turing's oracle machines [3]; the formulations of models and algorithms of hypercomputation have applied a wide spectrum of underlying theories [1,4,5]. It is precisely due to the existence of Turing's oracle machines that J. Copeland and D. Proudfoot introduced the term 'hypercomputation' by 1999 [6] for to replace the wrong expressions such as 'super-Turing computation', 'computing beyond Turing's limit', and 'breaking the Turing barrier', and similar.Recently Tien D. Kieu has proposed an quantum algorithm to solve the TM incomputable ‡ problem named Hilbert's tenth problem, using as physical referent the well known simple harmonic oscillator (SHO), which by effect of the second quantization has as associated dynamical algebra the Weyl-Heisenberg algebra denoted g W−H [8,9,10,11,12,13]. From the algebraic analysis of Kieu's hypercomputational quantum algorithm (KHQA), we have identified the underlying properties of the g W−H algebra which are necessary (but not sufficient) to guaranty KHQA works.…”

mentioning

confidence: 99%

Quantum hypercomputation based on the dynamical algebra

Sicard¹,

Ospina²,

Vélez³

2006

J. Phys. A: Math. Gen.

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Abstract. An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl-Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible representations with naturally defined generalized coherent states. We have selected the Lie algebra su(1, 1), due to that this algebra posses the necessary characteristics for to realize the hypercomputation and also due to that such algebra has been identified as the dynamical algebra associated to many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra su(1, 1), we presented an adaptations of KHQA over some concrete physical referents: the infinite square well, the infinite cylindrical well, the perturbed infinite cylindrical well, the Pöschl-Teller potentials, the Holstein-Primakoff system, and the Laguerre oscillator. We conclude that it is possible to have many physical systems within condensed matter and quantum optics on which it is possible to consider an implementation of KHQA. Quantum hypercomputation based on the dynamical algebra su(1, 1) 2 IntroductionThe hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine (TM) [1,2]. Starting from that seems to be the first published model of hypercomputation, which is called the Turing's oracle machines [3]; the formulations of models and algorithms of hypercomputation have applied a wide spectrum of underlying theories [1,4,5]. It is precisely due to the existence of Turing's oracle machines that J. Copeland and D. Proudfoot introduced the term 'hypercomputation' by 1999 [6] for to replace the wrong expressions such as 'super-Turing computation', 'computing beyond Turing's limit', and 'breaking the Turing barrier', and similar.Recently Tien D. Kieu has proposed an quantum algorithm to solve the TM incomputable ‡ problem named Hilbert's tenth problem, using as physical referent the well known simple harmonic oscillator (SHO), which by effect of the second quantization has as associated dynamical algebra the Weyl-Heisenberg algebra denoted g W−H [8,9,10,11,12,13]. From the algebraic analysis of Kieu's hypercomputational quantum algorithm (KHQA), we have identified the underlying properties of the g W−H algebra which are necessary (but not sufficient) to guaranty KHQA works. Such properties are that the dynamical algebra admits infinite-dimensional irreducible representations with naturally associated coherent states.The importance of KHQA inside the field of hypercomputation, at the same tenor that the importance of hypercomputation within the domain of computer science, can not be sub-estimated. This algorithm is a plausible candidate for a practical implementation of the hypercomputation, maybe within the scope of the quantum optics. The adaptation of KHQA to a new physical referents different to the harmonic oscillator, opens the possibility of analyze news viable alterna...

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Computing the non-computable

Cited by 42 publications

References 14 publications

Two-dimensional Ising model with competing interactions and its application to clusters and arrays ofπ-rings and adiabatic quantum computing

Two-dimensional Ising model with competing interactions and its application to clusters and arrays ofπ-rings and adiabatic quantum computing

Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

Quantum hypercomputation based on the dynamical algebra

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