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 2⁻ᶿ⁽ⁿᵋ⁾ and uses a qudit channel n(1 + Θ(√ε)) times, of which ε 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 √ε 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(√(ε log log 1/ε)) 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 small ε
