Chow$apos$s theorem and universal holonomic quantum computation

Abstract

A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to Holonomic Quantum Computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated to a non-Abelian gauge connection. When applied to the Optical Holonomic Computer, these conditions determine that the holonomy group of the two-qubit interaction model contains $SU(2) \times SU(2)$. In particular, a universal two-qubit logic gate is attainable for this model.