Extended Data Fig. 4: Detailed analysis of time-trace performance at the PMM sweet spot.

a,b, Signal-to-noise ratio (SNR_M, panel a) and parity lifetimes (panel b) as a function of readout amplitude. All data were taken with the same gate settings and 150-μs integration time as in Fig. 3d. The dotted grey line indicates the amplitude used for Figs. 3 and 4. Beyond a certain readout amplitude, the SNR decreases and the lifetimes become parity-dependent, consistent with drive-induced modulation along the common-mode axis δ that preferentially stabilizes the even ground state. This observation suggests that the system can be initialized in the even state by driving the middle resonator with a high amplitude. The time traces in panels a and b were averaged in time bins of 160 μs (instead of 150 μs) to ensure that the averaging time is an exact multiple of the sampling rate (50 kHz). c,d, SNR_M (c) and readout error (d) of the time trace presented in Fig. 3 versus integration time. The readout error was estimated as \([1-{{\rm{e}}}^{-{\tau }_{{\rm{bin}}}/{\tau }_{{\rm{avg}}}}{\rm{erf}}({{\rm{SNR}}}_{{\rm{M}}}({\tau }_{{\rm{bin}}})/\sqrt{2})]/2\) (ref. ²³), in which τ_bin is the integration time and τ_avg = 1.85 ± 0.03 ms. The dotted grey line indicates the integration time τ_bin = 150 μs used for all of the reported measurements unless otherwise specified. e, Histograms of dwell times in the even and odd parity states, normalized by total counts. The dotted lines show fits with the corresponding Poissonian probability densities \({{\rm{e}}}^{-t/{\tau }_{{\rm{o,e}}}}/{\tau }_{{\rm{o,e}}}\), confirming that the dwell times follow an exponential distribution. τ_o,e have been estimated as the mean of the dwell times in the odd and even parity states. Their uncertainty has been estimated as the standard deviation of the mean. f, Average switching time, τ_avg, estimated for varying integration time. As the integration time increases, the estimation of τ_avg is systematically biased towards higher values. See Methods section ‘Time traces measurement and analysis’ for a more detailed discussion. g, Power spectral density (PSD) of the time trace using an integration time of 10 μs. By fitting it with a Lorentzian model, we extract an average switching time of 1.51 ± 0.07 ms. The discrepancy between this estimate and those obtained with a hidden Markov model or exponential fitting further suggests that the long integration time needed to obtain high SNR could systematically bias the estimation of τ_avg. See Methods section ‘Time traces measurement and analysis’ for a more detailed discussion. h,i, Analysis of the time trace of Fig. 3c using the analysis methods and code adapted from ref. ²³. In panel h, the histogram is fit to a weighted sum of Gaussian distributions. The extracted SNR = d/(σ_e + σ_o) = 1.93 is comparable with the hidden Markov model analysis. In panel i, the dwell time distributions are extracted using a Gaussian mixture model and a thresholding algorithm. The parity lifetimes are then estimated by fitting the distributions with an exponential model. The estimates are comparable with those obtained with the hidden Markov model analysis.