Breeding Grid States From Schrödinger Cat States without Post-Selection

Grid (or comb) states are an interesting class of bosonic states introduced by Gottesman, Kitaev and Preskill to encode a qubit into an oscillator. A method to generate or `breed' a grid state from Schrödinger cat states using beam splitters and homodyne measurements is known but this method requires post-selection. We show how post-processing of the measurement data can be used to entirely remove the need for post-selection, making the scheme much more viable.
Bounding the asymptotic behavior of the breeding procedure is challenging. We show how a new class of approximate grid states that is invariant under the breeding operation can be used to obtain such bounds.
Finally, we demonstrate the efficacy of the method numerically.