Quantum Hypercomputation

Amit Hagar, Alexandre Korolev

A Sicard, J Ospina, M Vélez

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 , because this algebra possesses the necessary characteristics to realize the hypercomputation and also because such algebra has been identified as the dynamical algebra associated with many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra , we presented an adaptation 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 in which it is possible to consider an implementation of KHQA.

Gentian Kasa

The purpose of this thesis is to make an analysis of the concept of Hypercomputation and of some hypermachines. This thesis is separated in three main parts. We start in the first chapter with an analysis of the concept of Classical Computability with the Turing Machine and the Church-Turing thesis as a main reference and afterwards, in the second chapter, we continue with an analysis of hypercomputation and some hypermachines. Attention is given to the possible physical realization of these machines and their usefulness. In the third chapter a superficial introduction to Quantum Computing is made and a brief analysis to a quantum hypercomputational model, Tien D. Kieu's Adiabatic Quantum Computing.

Mario E. Vélez, Andrés Sicard, Juan Ospina

Hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. An adaptation of Tien D. Kieu¿s quantum hypercomputational algorithm is carried out for the dynamical algebra su(1, 1) of the Poschl-Teller potentials. The classically incomputable problem that is resolved with this hypercomputational algorithm is Hilbert¿s tenth problem. We indicated that an essential mathematical condition of these algorithms is the existence of infinitedimensional unitary irreducible representations of low dimensional dynamical algebras that allow the construction of coherent states of the Barut-Girardello type. In addition, we presented as a particular case of our hypercomputational algorithm on Poschl-Teller potentials, the hypercomputational algorithm on an infinite square well presented previously by the authors.