  A two-party quantum communication process with classical inputs and outcomes
can be simulated by replacing the quantum channel with a classical one. The
minimal amount of classical communication required to reproduce the statistics
of the quantum process is called its communication complexity. In the case of
many instances simulated in parallel, the minimal communication cost per
instance is called the asymptotic communication complexity. Previously, we
reduced the computation of the asymptotic communication complexity to a convex
minimization problem. In most cases, the objective function does not have an
explicit analytic form, as the function is defined as the maximum over an
infinite set of convex functions. Therefore, the overall problem takes the form
of a minimax problem and cannot directly be solved by standard optimization
methods. In this paper, we introduce a simple algorithm to compute the
asymptotic communication complexity. For some special cases with an analytic
objective function one can employ available convex-optimization libraries. In
the tested cases our method turned out to be notably faster. Finally, using our
method we obtain 1.238 bits as a lower bound on the asymptotic communication
complexity of a noiseless quantum channel with the capacity of 1 qubit. This
improves the previous bound of 1.208 bits.
