  We study analytically and numerically the behavior of quantum entanglement in
a quantum computer operating an efficient algorithm for quantum chaos. Our
results show that in an ideal algorithm the entanglement decays exponentially
with the diffusive relaxation rate induced by classical chaos. This decay goes
down to a residual level which drops exponentially with the number of qubits
n_q. Decoherence destroys the residual entanglement with a rate exponential in
n_q.
