  We calculate exact convergence times to reach random bipartite entanglement
for various random protocols. The eigenproblem of a Markovian chain governing
the process is mapped to a spin chain, thereby obtaining exact expression for
the gap of the Markov chain for any number of qubits. For protocols coupling
nearest neighbor qubits and CNOT gate the mapping goes to XYZ model while for
U(4) gate it goes to an integrable XY model. For coupling between a random pair
of qubits the mapping is to an integrable Lipkin-Meshkov-Glick model. In all
cases the gap scales inversely with the number of qubits, thereby improving on
a recent bound in [Phys.Rev.Lett. 98, 130502 (2007)].
