Numerical Simulations of a Possible Hypercomputational Quantum Algorithm

Sicard, Andrés; Ospina, Juan; Vélez, Mario

doi:10.1007/3-211-27389-1_65

Cited by 2 publications

(5 citation statements)

References 19 publications

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“…The adaptation of the KHQA for the case of the infinite square well (ISW) was realized by the present authors within a previous work [22,23]. At the present work we again establish that the ISW satisfy the mathematical structure given by ( 14), for a particular forms of f ISW y h ISW from which it is possible to construct the constitutive elements of the basic algebraic anatomy of the KHQA.…”

Section: The Infinite Square Wellsupporting

confidence: 68%

“…With base on (22), the algebra su(1, 1) admits an infinite-dimensional UIR over the space F ISW , which is given by…”

Section: The Infinite Square Wellmentioning

confidence: 99%

“…Due to the spectral structure of the ISW, the dynamical algebra associated with it, is the Lie algebra su(1, 1) [17] whose generators denoted K ISW + , K ISW − and K ISW 3 satisfy the commutation relations of (14b). With base on (22), the algebra su(1, 1) admits an infinite-dimensional UIR over the space F ISW , which is given by…”

Section: The Infinite Square Wellmentioning

confidence: 99%

“…where I v (x) is the modified Bessel function of the first kind. The corresponding probability density for the random variable n that results from (23) es…”

Section: The Infinite Square Wellmentioning

confidence: 99%

“…In the section 4 we note that the infinite cylindrical well and a modified cylindrical well also admit a realization of the su(1, 1) algebra. Moreover, based on the adaptation of KHQA that we have realized using the infinite square well [22,23], we show new adaptations of KHQA for some of the physical referents previously listed. Finally we present some conclusions.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Infinite Square Wellsupporting

confidence: 68%

“…With base on (22), the algebra su(1, 1) admits an infinite-dimensional UIR over the space F ISW , which is given by…”

Section: The Infinite Square Wellmentioning

confidence: 99%

Section: The Infinite Square Wellmentioning

confidence: 99%

“…where I v (x) is the modified Bessel function of the first kind. The corresponding probability density for the random variable n that results from (23) es…”

Section: The Infinite Square Wellmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quantum hypercomputation based on the dynamical algebra

Sicard¹,

Ospina²,

Vélez³

2006

J. Phys. A: Math. Gen.

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<title>A possible hypercomputational quantum algorithm</title>

2005

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The term 'hypermachine' denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically non-computable decision problem, Hilbert's tenth problem; more specifically, we present a possible hypercomputation model based on quantum computation. Our algorithm is inspired by the one proposed by Tien D. Kieu, but we have selected the infinite square well instead of the (one-dimensional) simple harmonic oscillator as the underlying physical system. Our model exploits the quantum adiabatic process and the characteristics of the representation of the dynamical Lie algebra su(1, 1) associated to the infinite square well.

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Numerical Simulations of a Possible Hypercomputational Quantum Algorithm

Cited by 2 publications

References 19 publications

Quantum hypercomputation based on the dynamical algebra

Quantum hypercomputation based on the dynamical algebra

<title>A possible hypercomputational quantum algorithm</title>

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