  The $N$ vertices of a quantum random graph are each a circle independently
punctured at Poisson points of arrivals, with parallel connections derived
through for each pair of these punctured circles by yet another independent
Poisson process. Considering these graphs at their critical parameters, we show
that the joint law of the re-scaled by $N^{2/3}$ and ordered sizes of their
connected components, converges to that of the ordered lengths of excursions
above zero for a reflected Brownian motion with drift. Thereby, this work forms
the first example of an inhomogeneous random graph, beyond the case of
effectively rank-1 models, which is rigorously shown to be in the
Erd\H{o}s-R\'{e}nyi graphs universality class in terms of Aldous's results.
