  Gaussian states and measurements collectively are not powerful-enough
resources for quantum computing, as any Gaussian dynamics can be simulated
efficiently, classically. However, it is known that any one non-Gaussian
resource -- either a state, a unitary operation, or a measurement -- together
with Gaussian unitaries, makes for universal quantum resources. Photon number
resolving (PNR) detection, a readily-realizable non-Gaussian measurement, has
been a popular tool to try and engineer non-Gaussian states for universal
quantum processing. In this paper, we consider PNR detection of a subset of the
modes of a zero-mean pure multi-mode Gaussian state as a means to herald a
target non-Gaussian state on the undetected modes. This is motivated from the
ease of scalable preparation of Gaussian states that have zero mean, using
squeezed vacuum and passive linear optics. We calculate upper bounds on the
fidelity between the actual heralded state and the target state. We find that
this fidelity upper bound is $1/2$ when the target state is a multi-mode
coherent cat-basis cluster state, a resource sufficient for universal quantum
computing. This proves that there exist non-Gaussian states that are not
producible by this method. Our fidelity upper bound is a simple expression that
depends only on the target state represented in the photon-number basis, which
could be applied to other non-Gaussian states of interest.
