Abstract
Long time behavior of a unitary quantum gate U, acting sequentially on two subsystems of dimension N each, is investigated. We derive an expression describing an arbitrary iteration of a two-qubit gate making use of a link to the dynamics of a free particle in a 3D billiard. Due to ergodicity of such a dynamics an average along a trajectory V^t stemming from a generic two-qubit gate V in the canonical form tends for a large t to the average over an ensemble of random unitary gates distributed according to the flat measure in the Weyl chamber - the minimal 3D set containing points from all orbits of locally equivalent gates. Furthermore, we show that for a large dimension N the mean entanglement entropy averaged along a generic trajectory coincides with the average over the ensemble of random unitary matrices distributed according to the Haar measure on U(N²).