  Quantum fingerprinting reduces communication complexity of determination
whether two $n$-bit long inputs are equal or different in the simultaneous
message passing model. Here we quantify the advantage of quantum fingerprinting
over classical protocols when communication is carried out using optical
signals with limited power and unrestricted bandwidth propagating over additive
white Gaussian noise (AWGN) channels with power spectral density (PSD) much
less than one photon per unit time and unit bandwidth. We identify a noise
parameter whose order of magnitude separates near-noiseless quantum
fingerprinting, with signal duration effectively independent of $n$, from a
regime where the impact of AWGN is significant. In the latter case the signal
duration is found to scale as $O(\sqrt{n})$, analogously to classical
fingerprinting. However, the dependence of the signal duration on the AWGN PSD
is starkly distinct, leading to quantum advantage in the form of a reduced
multiplicative factor in $O(\sqrt{n})$ scaling.
