Quantum hypercomputation based on the dynamical algebra

Abstract

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...

On SU(1,1) intelligent coherent states

“…The SU(1,1) Lie algebra is used in studying the adaptation of Kieu's hypercomputational quantum algorithm [93] and for quantum computation and tomography [94]. The SU(1,1) dynamical algebra is selected, because it possesses the necessary characteristics in realizing the hypercomputational quantum algorithm.…”

“…In addition, it admits different kinds of CSs and various kinds of representations. Some realizations of SU(1,1) Lie algebra such as Holstein-Primakoff and one-mode nonlinear realization are used for studying the quantum hypercomputation [93]. Moreover, the approximation correspondence between the SU(1,1) Lie algebra and finite elements of quantum gates are derived [94].…”

Numerical Simulations of a Possible Hypercomputational Quantum Algorithm

“…Therefore, in principle, it is possible to select one of these referents as an underlying physical system of our hypercomputational quantum algorithm. 23…”