Extended Data Fig. 3: Noise model and comparison with experiment (extended figure 3).

a) Trap frequency measured at a time delayed from a 50 Hz line trigger. The color shows the probability of finding the ion in \({\left|0\right\rangle }_{S}\) after the frequency calibration experiment described in the text. The solid line is a fit using the five lowest harmonics of 50 Hz. b) A measurement of motional coherence of the state \((\left|0\right\rangle +\left|1\right\rangle )/\sqrt{2}\), taken using a Ramsey sequence. An exponential fit yields a coherence time of 16.4(9) ms. The solid line is the result of the Monte-Carlo wavefunction simulation based on the noise parameters given in the text. c) Decay simulation of a GKP \(\left|{1}_{L}\right\rangle\) logical state under the independent action of each error channel: Markovian dephasing (orange), 50 Hz noise (green) and heating (blue) with (solid lines) and without (dashed lines) stabilization. For each error channel we optimise the frequency at which we apply the error correction. d-e) Comparison of data with simulation for the time evolution of logical readouts \(\left\langle {X}_{L}^{f}\right\rangle\) (orange), \(\left\langle {Y}_{L}^{f}\right\rangle\) (green) and \(\left\langle {Z}_{L}^{f}\right\rangle\) (blue) with stabilization (solid lines) and without stabilization (dashed lines) for both the d) square and e) hexagonal finite-GKP encoding. The simulation reproduces qualitatively the experimental data for both. The gray curves represent the coherence of the \((\left|0\right\rangle +\left|1\right\rangle )/\sqrt{2}\) Fock state superposition extracted from b).