Efficient universal quantum computation with auxiliary Hilbert space

We propose a scheme to construct the efficient universal quantum circuit for qubit systems with the assistance of possibly available auxiliary Hilbert spaces. An elementary two-ququart gate, termed the controlled-double-NOT gate, is proposed first in ququart (four-level) systems, and its physical implementation is illustrated in the four-dimensional Hilbert spaces built by the path and polarization states of photons. Then an efficient universal quantum circuit for ququart systems is constructed using the gate and the quantum Shannon decomposition method. By introducing auxiliary two-dimensional Hilbert spaces, the universal quantum circuit for qubit systems is finally achieved using the result obtained in ququart systems with the lowest complexity

Exact solutions for a family of spin-boson systems

We obtain the exact solutions for a family of spin-boson systems. This is achieved through application of the representation theory for polynomial deformations of the $su(2)$ Lie algebra. We demonstrate that the family of Hamiltonians includes, as special cases, known physical models which are the two-site Bose-Hubbard model, the Lipkin-Meshkov-Glick model, the molecular asymmetric rigid rotor, the Tavis-Cummings model, and a two-mode generalisation of the Tavis-Cummings model.Comment: LaTex 15 pages. To appear in Nonlinearit

Polynomial algebras and exact solutions of general quantum non-linear optical models I: Two-mode boson systems

We introduce higher order polynomial deformations of $A_1$ Lie algebra. We construct their unitary representations and the corresponding single-variable differential operator realizations. We then use the results to obtain exact (Bethe ansatz) solutions to a class of 2-mode boson systems, including the Boson-Einstein Condensate models as special cases. Up to an overall factor, the eigenfunctions of the 2-mode boson systems are given by polynomials whose roots are solutions of the associated Bethe ansatz equations. The corresponding eigenvalues are expressed in terms of these roots. We also establish the spectral equivalence between the BEC models and certain quasi-exactly solvable Sch\"ordinger potentials.Comment: 20 pages, final version to appear in J. Phys. A: Math. Theor

Quasi-exactly solvable models derived from the quasi-Gaudin algebra

The quasi-Gaudin algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation of the quasi-Gaudin algebra, we obtain a class of bosonic models which exhibit this curious property. These models have the notable feature that they do not preserve U(1) symmetry, which is typically associated with a non-conservation of particle number. An exact solution for the eigenvalues within the quasi-exactly solvable sector is obtained via the algebraic Bethe ansatz formalism

Polynomial algebras and exact solutions of general quantum nonlinear optical models: II. Multi-mode boson systems

We present higher order polynomial algebras which are the dynamical symmetry algebras of a wide class of multi-mode boson systems in nonlinear optics and laser physics. We construct their unitary representations and the corresponding single-variable differential operator realizations. We then use the results to obtain exact (Bethe ansatz) solutions to the multi-mode boson systems, including the Bose-Einstein condensate models as the special cases