Decoherence in Holonomic Quantum Computation

Geometric phases are an interesting field of research in quantum mechanics. Recently both abelian and nonabelian geometric phases have been proposed as a useful resource for the experimental implementation of quantum computation. In this paper we focus on a particular physical model and study the effect of a bosonic bath on a class of holonomic transformations. We write a general master equation for time-dependent Hamiltonians and derive analytical and numerical solutions for the system considered. The fidelity is analyzed in the adiabatic and nonadiabatic regime. We also determine an optimal finite operation time for this class of gates.