Encoding quantum information in a harmonic oscillator offers a resource efficient method for quantum error correction, compared to the use of multiple two-level systems. The Gottesman-Kitaev-Preskill (GKP) encoding  is particularly promising and has recently been realized in both trapped ions [2, 3] and superconducting microwave cavities .
State preparation, readout, single qubit rotations and error correction have been achieved in both of these architectures. I will describe work towards logical entangling gates between two GKP qubits prepared in the motional modes of two atomic ions trapped in close proximity. The modes are coupled via the Coulomb interaction, which approximates a beam splitter. Combined with single mode squeezing operations this beam splitter interaction can realize universal entangling gates between the two GKP states . In theoretical work, we have investigated this gate for experimentally realistic parameters and finite energy GKP states. In parallel, we are developing an apparatus for these experiments, including fabrication of a suitable ion trap and design and implementation of optical addressing with tightly focused laser beams.
 Daniel Gottesman, Alexei Kitaev and John Preskill, Physical Review A 64, 012310 (2001)
 Christa Flühmann et al., Nature 566, 368-372 (2019)
 Brennan de Neeve et al., Nature Physics 18, 296-300 (2022)
 Philippe Campagne-Ibarcq et al., Nature 584, 368-372 (2020)
 Ilan Tzitrin et al., Physical Review A 101, 032315 (2020)
|Presenter name||Martin Wagener|
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