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 noiseless qudit channels (where d is arbitrary), our main result is a simulation method that fails with probability less than exp(-Theta(n epsilon)) and uses a qudit channel n(1 + Theta (sqrt(epsilon))) 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. Surprisingly, this outperforms the best known overhead of 1 + O(sqrt(epsilon log log 1/epsilon)) in the corresponding classical model, which is also conjectured to be optimal [Haeupler, FOCS'14]. Our work also 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. This is joint work with Debbie Leung, Ashwin Nayak, Ala Shayeghi, Penghui Yao and Nengkun Yu.