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A distributed quantum computing system requires a quantum communication channel between spatially separated processing units. In superconducting circuits, such a channel can be realized by using propagating microwave photons to encode and transfer quantum information between an emitter and a receiver node. Here we experimentally demonstrate a superconducting circuit that deterministically transfers the state of a data qubit into a propagating microwave mode, with a process fidelity of 94.5%. We use a time-varying parametric drive to shape the temporal profile of the propagating mode to be time-symmetric and with constant phase, so that reabsorption by the receiving processor can be implemented as a time-reversed version of the emission. We demonstrate a self-calibrating routine to correct for time-dependent shifts of the emitted frequencies due to the modulation of the parametric drive. Our work provides a reliable method to implement high-fidelity quantum state transfer and remote entanglement operations in a distributed quantum computing network.