INTEGRABILITY, ENTROPY AND QUANTUM COMPUTATION

The important requirements are stated for the success of quantum computation. These requirements involve coherent preserving Hamiltonians as well as exact integrability of the corresponding Feynman path integrals. Also we explain the role of metric entropy in dynamical evolutionary system and outline some of the open problems in the design of quantum computational systems. Finally, we observe that unless we understand quantum nondemolition measurements, quantum integrability, quantum chaos and the direction of time arrow, the quantum control and computational paradigms will remain elusive and the design of systems based on quantum dynamical evolution may not be feasible.

Exact Integrability of the su(n) Hubbard Model

The bosonic su (n) Hubbard model was recently introduced. The model was shown to be integrable in one dimension by exhibiting the infinite set of conserved quantities. I derive the R-matrix and use it to show that the conserved charges commute among themselves. This new matrix is a nonadditive solution of the Yang–Baxter equation. Some properties of this matrix are derived.

EXACT BETHE ANSATZ SOLUTION OF NON-ULTRALOCAL QUANTUM mKdV MODEL

A lattice regularized Lax operator for the non-ultralocal modified Korteweg de Vries (mKdV) equation is proposed at the quantum level with the basic operators satisfying a q-deformed braided algebra. From the associated quantum R and Z-matrices the exact integrability of the model is proved through the braided quantum Yang-Baxter equation which is a suitably generalized equation for the non-ultralocal models. Using the algebraic Bethe ansatz the eigenvalue problem of the quantum mKdV model is exactly solved and its connection with the spin-1/2 XXZ chain is established, facilitating the investigation of the corresponding conformal properties.

ON EXACT INTEGRABILITY OF 2-D POINCARÉ GRAVITY

We consider the 2-D Poincaré gravity and show its exact integrability. The choice of the gauge is discussed. The Euclidean solutions on compact closed differential manifolds are studied.

THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS IN (1+1)d QUANTUM FIELD THEORY FROM FINITE SIZE EFFECTS IN CONFORMAL THEORIES

Using the finite-size effects, the scaling dimensions and correlation functions of the main operators in continuous and lattice models of 1d spinless Bose-gas with pairwise interaction of rather general form are obtained. The long-wave properties of these systems can be described by the Gaussian model with central charge c=1. The disorder operators of the extended Gaussian model are found to correspond to some nonlocal operators in the XXZ Heisenberg antiferromagnet. This same approach is applicable to fermionic systems. Scaling dimensions of operators and correlation functions in the systems of interacting Fermi-particles are obtained. We present a universal treatment for 1d systems of different kinds which is independent of the exact integrability and which gives universal expressions for critical exponents through the thermodynamic characteristics of the system.