Extended Data Fig. 7: Expected qubit coherence from extracted noise parameters.

We simulate the expected CPMG-1 decay using the filter formalism as detailed in the Methods section, using the noise power parameters as extracted from the CPMG experiment displayed in Fig. 5 of the main text: \(\sqrt{{S}_{V}}=24.7\,\mu {\rm{V}}/\sqrt{\,{{\mbox{Hz}}}\,},{\gamma }_{{{{\rm{Ge}}}}{{\mbox{-}}}73}=1.48 \;{{\rm{MHz}}/{\mathrm{T}}}\), \({S}_{0,{{{\rm{hf}}}}}=2.5\times 1{0}^{12}{\cos }^{2}({\theta }_{{f}_{{{{\rm{Q}}}}2}}),{\sigma }_{{{{\rm{Ge}}}}{{\mbox{-}}}73}\) \(=17 \;{\rm{kHz}}\) for ϕ_B = 0°, and σ_Ge-73 = 9 kHz for ϕ_B = −105°. We then fit the simulated decay using the same procedure as used in Fig. 3f of the main text to extract the envelope \({T}_{2}^{\,{\rm{H}}}\). Markers indicate experimental data (reproduced from Fig. 3) and the solid lines correspond to the envelope decay time as predicted by the model. The excellent agreement between the simulation and data, without the need for any fitting parameters, confirms our understanding of the system. The 1/f_Q2 decay of the envelope coherence can be explained by an effective voltage noise on plunger gate P2, while the low-B drop-off is caused by the finite spread of the ⁷³Ge precession frequencies. The shaded area indicates the uncertainty in \({T}_{\,{\rm{H}}}^{2}\), given an uncertainty of ± 20 μT in the z-component of the magnetic field. For very small B, this yields a significant uncertainty in θ_B, thus complicating an accurate prediction of \({T}_{\,{\rm{H}}}^{2}\).