  Hybrid quantum information processing combines the advantages of discrete and
continues variable protocols by realizing protocols consisting of photon
counting and homodyne measurements. However, the mode structure of pulsed
sources and the properties of the detection schemes often require the use
optical filters in order to combine both detection methods in a common
experiment. This limits the efficiency and the overall achievable squeezing of
the experiment. In our work, we use photon subtraction to implement the
distillation of pulsed squeezed states originating from a genuinely spatially
and temporally single-mode parametric down-conversion source in non-linear
waveguides. Due to the distillation, we witness an improvement of
$0.17~\mathrm{dB}$ from an initial squeezing value of $-1.648 \pm
0.002~\mathrm{dB}$, while achieving a purity of $0.58$, and confirm the
non-Gaussianity of the distilled state via the higher-order cumulants. With
this, we demonstrate the source's suitability for scalable hybrid quantum
network applications with pulsed quantum light.
