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An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) ispresented. The method that was used was to replace the Weyl-Heisenberg algebraby other dynamical algebra of low dimension that admits infinite-dimensionalirreducible representations with naturally defined generalized coherent states.We have selected the Lie algebra $\mathfrak{su}(1,1)$, due to that this algebraposses the necessary characteristics for to realize the hypercomputation andalso due to that such algebra has been identified as the dynamical algebraassociated to many relatively simple quantum systems. In addition to analgebraic adaptation of KHQA over the algebra $\mathfrak{su}(1,1)$, wepresented an adaptations of KHQA over some concrete physical referents: theinfinite square well, the infinite cylindrical well, the perturbed infinitecylindrical well, the P{\"o}sch-Teller potentials, the Holstein-Primakoffsystem, and the Laguerre oscillator. We conclude that it is possible to havemany physical systems within condensed matter and quantum optics on which it ispossible to consider an implementation of KHQA.Comment: 25 pages, 1 figure, conclusions rewritten, typing and language errors  corrected and latex format changed minor changes elsewhere and

Quantum hypercomputation based on the dynamical algebra