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Bosonic codes provide a hardware-efficient option for error correction, but fault-tolerance further requires that the available gate interactions be compatible with the code. A promising bosonic code is the Gottesman-Kitaev-Preskill (GKP) code [1] for which the beamsplitter, a linear coupling between two bosonic modes, is fault-tolerant [1–5]. Entangling operations between GKP qubits have so far been realized only indirectly, through ancilla-mediated two-qubit gates rather than a direct coupling between the bosonic modes. Such schemes are not fault-tolerant: errors on the ancilla can propagate into the encoded qubits in uncorrectable ways. Here [6], using two motional modes of a trapped ion, we demonstrate the generation of entangled states of GKP qubits by interfering two qunaught states [4], which have a grid structure but carry no logical information, on a beamsplitter. We generate all four Bell states with an average fidelity of 69%, and subsequently demonstrate an extension of the entangled state lifetime through the use of quantum error correction. These results complete the set of Gaussian operations required for quantum computing with GKP codes [3] and enable explorations of multi-mode bosonic encodings [7] as well as fundamental tests of information channels [8].

[1] Gottesman, D., Kitaev, A. & Preskill, J. Encoding a qubit in an oscillator. Phys. Rev. A 64, 012310 (2001).

[2] Glancy, S. & Knill, E. Error analysis for encoding a qubit in an oscillator. Phys. Rev. A 73, 012325 (2006).

[3] Menicucci, N. C. Fault-tolerant measurement-based quantum computing with continuous-variable cluster states. Phys. Rev. Lett. 112, 120504 (2014).

[4] Walshe, B. W., Baragiola, B. Q., Alexander, R. N. & Menicucci, N. C. Continuous-variable gate teleportation and bosonic-code error correction. Phys. Rev. A 102, 062411 (2020).

[5] Shaw, M. H., Doherty, A. C. & Grimsmo, A. L. Logical gates and read-out of superconducting Gottesman-Kitaev-Preskill qubits. Preprint at arXiv:2403.02396 (2024).

[6] Fontboté-Schmidt, M. et al. In preparation.

[7] Royer, B., Singh, S. & Girvin, S. M. Encoding qubits in multimode grid states. PRX Quantum 3, 010335 (2022).

[8] Wang, Z. & Jiang, L. Passive environment-assisted quantum communication with GKP states. Phys. Rev. X 15, 021003 (2025).