The hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. Several numerical simulations of a possible hypercomputational algorithm based on quantum computations previously constructed by the authors are presented. The hypercomputability of our algorithm is based on the fact that this algorithm could solve a classically noncomputable decision problem, the Hilbert's tenth problem. The numerical simulations were realized for three types of Diophantine equations: with and without solutions in non-negative integers, and without solutions by way of various traditional mathematical packages.

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...

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.