Linear-optical generation of hybrid GKP entanglement from small-amplitude cat states

Abstract

Hybrid bosonic codes combining bosonic codes with photon states offer a promising pathway for fault-tolerant quantum computation. However, the efficient generation of such states in optical setups remains technically challenging due to the requirement for complex non-Gaussian resources. In this paper, we propose a novel scheme to efficiently generate hybrid entangled states between a GKP qubit and a photon-number state using small-amplitude cat states as the primary resource. We apply a breeding process using small-amplitude cat states to increase the non-Gaussianity of the input states. This method requires only linear optical elements and homodyne measurements. Furthermore, we demonstrate that this protocol can be extended to generate hybrid qudit states. This scheme has the potential to provide a resource-efficient and experimentally attractive route toward implementing hybrid quantum error correction.

Figures (5)

• Figure 1: Schematic of the proposed protocol. Inputs are initialized as cat states and a vacuum state. The circuit employs beam splitters, a displacement operation and a homodyne measurement ($p=0$) to generate the target hybrid entangled state.
• Figure 2: Fidelity between the ideal target state $\ket{\psi_o}$ and the generated state $\ket{\psi'}$. The optimal value appears at a finite amplitude, $\alpha \approx 0.455$, resulting from trade-off between the small-amplitude approximation and the overlap with the target state, rather than $\alpha \to 0$.
• Figure 3: We plot the average fidelity and success probability against the homodyne acceptance window $v_{up}$. This reveals a clear trade-off between the two metrics. The dashed curve indicates the average fidelity $F(v_{up})$. The dotted curve traces the success probability $P(v_{up})$. The resource state amplitude is fixed at the optimal value of $\alpha \approx 0.455$.
• Figure 4: Enhanced protocol via cat breeding. (a) Optical setup. We use a high non-Gaussianity state $\ket{\tilde{0}_{L}^{(j)}}$ as the input state instead of a single photon. (b) Equivalent circuit using the breeding protocol with all input states as single photons. (c) Wigner function of the output hybrid state. It demonstrates the enhanced non-Gaussianity achieved by breeding.
• Figure 5: Optical setup for generating a hybrid qutrit state. The input is the hybrid entangled state given by Eq. (\ref{['Eq:OutStateBeta']}). The hybrid qutrit state is generated when the homodyne measurement result is $p=0$ and the photon detector (PD) projects onto $_2\bra{0}$.