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.