  Calculating the capacity of interference channels is a notorious open problem
in classical information theory. Such channels have two senders and two
receivers, and each sender would like to communicate with a partner receiver.
The capacity of such channels is known exactly in the settings of "very strong"
and "strong" interference, while the Han-Kobayashi coding strategy gives the
best known achievable rate region in the general case. Here, we introduce and
study the quantum interference channel, a natural generalization of the
interference channel to the setting of quantum information theory. We restrict
ourselves for the most part to channels with two classical inputs and two
quantum outputs in order to simplify the presentation of our results (though
generalizations of our results to channels with quantum inputs are
straightforward). We are able to determine the exact classical capacity of this
channel in the settings of "very strong" and "strong" interference, by
exploiting Winter's successive decoding strategy and a novel two-sender quantum
simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi
strategy is achievable with Holevo information rates, up to a conjecture
regarding the existence of a three-sender quantum simultaneous decoder. This
conjecture holds for a special class of quantum multiple access channels with
average output states that commute, and we discuss some other variations of the
conjecture that hold. Finally, we detail a connection between the quantum
interference channel and prior work on the capacity of bipartite unitary gates.
