Grid (or comb) states are an interesting class of bosonic states introduced by Gottesman, Kitaev, and Preskill [D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001)] to encode a qubit into an oscillator. A method to generate or ``breed'' a grid state from Schr\"odinger cat states using beam splitters and homodyne measurements is known [H. M. Vasconcelos, L. Sanz, and S. Glancy, Opt. Lett. 35, 3261 (2010)], but this method requires postselection. In this paper we show how postprocessing of the measurement data can be used to entirely remove the need for postselection, making the scheme much more viable. We bound the asymptotic behavior of the breeding procedure and demonstrate the efficacy of the method numerically.
Homodyne breeding is a promising optical route for generating large cat states and Gottesman--Kitaev--Preskill (GKP) grid states, using readily available squeezed-cat states as inputs. We analyze the performance of protocols, using squeezed single-photon seeds,
S^(r)∣1⟩, as squeezed-cat resources and propagating them through the same cat-to-GKP breeding pipeline. For a symmetric tree with
N=2m1 seeds, the heralded resource state reduces to a state with a polynomial-Gaussian wavefunction controlled by a single effective width, achieving near-unit fidelity with the desired cat resource state. Our analysis yields closed-form even-cat fidelities and an inverse-design rule for the seed squeezing. We apply these resources to GKP breeding, and find a large overlap with a practical GKP state. We also investigate the effects of practical imperfections, including finite homodyne acceptance windows and photon loss. This study shows that squeezed single-photon inputs provide a viable resource assignment for homodyne cat and GKP breeding, replacing the demanding requirement for high-resolution photon-number-resolving heralding with an on-demand single-photon source.