Analysis of Optical GKP State Generation Using Shaped Free Electrons

The Gottesman-Kitaev-Preskill (GKP) bosonic error-correcting codes are the major candidates to realize fault-tolerant quantum computing and communication with bosonic systems. Especially, the correction capabilities of the optical GKP states may combat fiber attenuation to efficiently produce high-fidelity entanglement over long distances. However, due to weak optical nonlinearities and the absence of efficient and high-fidelity microwave-to-optical transduction,  it remains highly challenging to produce the necessary GKP states in the optical domain scalably.

A recent proposal [Dahan et al., PRX 13, 031001 (2023)] considers the use of a shaped free electron to rapidly impart nonlinearities onto the photonic mode, generating a GKP state upon post-selection. We seek to understand and improve this approach in a few ways. To better represent the noise on the electron state, we rigorously describe the electronic wavefunction as a continuous-variable system in time-frequency space. Furthermore, we use the electron-photon interaction to produce more general states, such as the GKP qu-naught state, which can be used to produce a GKP Bell pair with a balanced beamsplitter. Finally, we remove the electron measurement by considering alternate state preparation procedures, reducing the experimental overhead and improving the post-selection.