Quantum gates and their coexisting geometric phases

Geometric phases arise naturally in a variety of quantum systems with observable consequences. They also arise in quantum computations when dressed states are used in gating operations. Here we show how they arise in these gating operations and how one may take advantage of the dressed states producing them. Specifically, we show that for a given, but arbitrary Hamiltonian, and at an arbitrary time τ, there always exists a set of dressed states such that a given gate operation can be performed by the Hamiltonian up to a phase φ. The phase is a sum of a dynamical phase and a geometric phase. We illustrate the dressed phase for several systems.