Finite energy corrections for correlated readout of a GKP Bell state in a single trapped ion

Bosonic codes such as the Gottesman-Kitaev-Preskill (GKP) code [1] provide an alternative approach for quantum error correction by using the extended Hilbert space of the continuous degrees of freedom to redundantly store information. Implementation of the GKP code in experimental platforms diverge from the ideal code due to their need to work with finite energy states. To cope with this adjustment, error correction protocols have been proposed and experimentally demonstrated [2,3], which maintain the finite energy of the states involved. Here we extend these techniques to demonstrate the state preparation of two GKP states in the radial modes of a single ion. We show that the methods for finite energy readout used for a single mode can be used in these larger systems, including to read out correlations. We also present work towards the implementation of two-qubit gates on error-corrected qubits using finite-energy methods [4], as well as the generation of a GKP logical Bell state by using grid states that contain no logical information and using a beamsplitter [5]. Using the finite energy correlated readout for quantum state tomography, we show measurements of the fidelity of our logical Bell state.



[1] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001)

[2] P. Campagne-Ibarcq et. al, Nature 584, 368 (2020).

[3] B. de Neeve, TL. Nguyen, T. Behrle, et al. Nat. Phys. 18, 296-300 (2022)

[4] I. Rojkov et. al. Phys. Rev. Lett. 133, 100601 (2024)

[5] B.W. Walshe, B.Q. Baragiola, R.N. Alexander, N.C. Menicucci, Phys. Rev. A 102, 062411 (2020)