It is now well-established that photonic systems can exhibit topological energy bands; similar to their electronic counterparts, this leads to the formation of chiral edge modes which can be used to transmit light in a manner that is protected against back-scattering. While it is understood how classical signals can propagate under these conditions, it is an outstanding important question how the quantum vacuum fluctuations of the electromagnetic field get modified in the presence of a topological band structure. We address this challenge by exploring a setting where a non-zero topological invariant guarantees the presence of a parametrically-unstable chiral edge mode in a system with boundaries, even though there are no bulk-mode instabilities. We show that one can exploit this to realize a topologically protected, quantum-limited travelling-wave parametric amplifier. The device is naturally protected both against internal losses and back-scattering; the latter feature is in stark contrast to standard travelling wave amplifiers. This adds a new example to the list of potential quantum devices that profit from topological transport.
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