  We consider the problem of implementing two-party interactive quantum
communication over noisy channels, a necessary endeavor if we wish to fully
reap quantum advantages for communication. For an arbitrary protocol with $n$
messages, designed for a noiseless qudit channel over a $\mathrm{poly}(n)$ size
alphabet, our main result is a simulation method that fails with probability
less than $2^{-\Theta(n\epsilon)}$ and uses a qudit channel over the same
alphabet $n\left(1+\Theta \left(\sqrt{\epsilon}\right)\right)$ times, of which
an $\epsilon$ fraction can be corrupted adversarially. The simulation is thus
capacity achieving to leading order, and we conjecture that it is optimal up to
a constant factor in the $\sqrt{\epsilon}$ term. Furthermore, the simulation is
in a model that does not require pre-shared resources such as randomness or
entanglement between the communicating parties. Our work improves over the best
previously known quantum result where the overhead is a non-explicit large
constant [Brassard et al., FOCS'14] for low $\epsilon$.
