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The discrete memoryless interference channel is modelled as a conditional probability distribution with two outputs depending on two inputs and has widespread applications in practical communication scenarios. In this paper, we introduce and study the quantum interference channel, a generalization of a two-input, two-output memoryless channel to the setting of quantum Shannon theory. We discuss three different coding strategies and obtain corresponding achievable rate regions for quantum interference channels. We calculate the capacity regions in the special cases of "very strong" and "strong" interference. The achievability proof in the case of "strong" interference exploits a novel quantum simultaneous decoder for two-sender quantum multiple access channels. We formulate a conjecture regarding the existence of a quantum simultaneous decoder in the three-sender case and use it to state the rates achievable by a quantum Han-Kobayashi strategy.