Performance analysis of cat and GKP breeding protocols with squeezed single-photon input
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, $\hat{S}(r)\ket{1}$, as squeezed-cat resources and propagating them through the same cat-to-GKP breeding pipeline. For a symmetric tree with $N=2^{m_1}$ 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.