Performance analysis of cat and GKP breeding protocols with squeezed single-photon input

Abstract

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.

The paper asks how squeezed single photons, $\hat S(r)|1\rangle$, perform as substitutes for squeezed-cat inputs in established optical cat-to-GKP homodyne-breeding protocols, avoiding high-photon-number-resolving heralding. For a symmetric balanced tree with $N=2^{m_1}$ seeds and zero homodyne outcomes, the conditional cat-resource wavefunction is derived as $\psi_N(x)\propto x^N e^{-x^2/(2\sigma_{\rm eff})}$, with $\sigma_{\rm eff}=e^{2r}$. Gaussian-moment calculations give closed-form even-cat fidelities and a one-parameter optimization of seed squeezing. At fidelity $\geq0.99$, the largest target amplitudes are approximately $1.65,2.38,3.31,4.65$ for $N=2,4,8,16$. One-step GKP breeding consumes two such resources. All tested ideal outputs have optimized overlap above $0.99$ with a finite-energy qunaught reference. Their symmetric stabilizer-based effective squeezing ranges from $3.30$ to $7.53\,\mathrm{dB}$ for $4$–$32$ seeds; increasing resource size mainly improves one quadrature. Additional GKP-breeding layers extend the comb and redistribute quadrature squeezing while doubling seed cost per layer. An analytic outcome-polynomial treatment evaluates finite-window probabilities and averaged cat fidelities without sampling. At acceptance half-width $w\simeq1$, fidelities remain above $0.9$, but acceptance falls from $0.42$ at $N=2$ to $1.18\times10^{-6}$ at $N=16$. Loss simulations show increasing fragility with resource size; below approximately $95\%$ transmissivity, $N=8$ outperforms $N=16$ in symmetric GKP squeezing. Limitations include power-of-two trees, squeezing below the cited $9.75\,\mathrm{dB}$ fault-tolerance benchmark, and no complete generation-rate model. Finite-window GKP-fusion performance is not established; quoted GKP squeezing uses ideal conditioning.