Quantum hypercomputation based on the dynamical algebra su(1,1)

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 $\mathfrak{su}(1,1)$, 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 $\mathfrak{su}(1,1)$, 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 P{ö}sch-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.

25 pages, 1 figure, conclusions rewritten, typing and language errors corrected and latex format changed minor changes elsewhere and l

References in corpus (8)

• A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
• Temporally stable coherent states for infinite well and Pöschl-Teller potentials
• Adiabatic Theorem without a Gap Condition
• Entangled SU(2) and SU(1,1) coherent states
• Exponential and Laguerre Squeezed States for su(1,1) Algebra and Calogero-Sutherland Model
• Two-mode Nonlinear Coherent States
• Coherent States for Generalized Laguerre Functions
• Meixner functions and polynomials related to Lie algebra representations