Our model assumes that an exponential decay well approximates a relaxation time experiment. However, this assumption breaks down in the presence of strong coupling between the qubit and a TLS, which induces oscillations in the excited-state return probability. If the oscillation frequency is known, the protocol can be easily adapted by sampling at the maxima (or minima) of these oscillations. Otherwise, the model must be extended for multiparameter Bayesian estimation [34, 49]. Such strong coupling regimes have not been observed in ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$ during our experiments.

A gamma distribution models the statistics of the sum of independent decay processes, where the random variable represents the total decay time. In that sense, it is naturally related to decay dynamics. However, in our case, we measure the state $m$ rather than the actual decay time $\tau$. As a result, the relevant statistics are binomial rather than the exponential distribution typically associated with lifetime studies. Therefore, the appearance of the gamma distribution in this work is not due to physical origins, but rather because it closely resembles the true posterior.

Thus for adaptive methods, the Shannon information is rarely used in practice, as it tends to be computationally intractable.

For $(\alpha =\beta =0)$, we have $\mathrm{E}[{\mathrm{\Gamma }}_1|m_i,{\nu }_i]=k/\theta$ and $\mathrm{E}[{\mathrm{\Gamma }}_1^2|m_i,{\nu }_i]=(k_i+k_i^2)/{\theta }_i^2$ (see Supplemental Material [44] for the case $\alpha$, $\beta \ne 0$).

We perform a numerical approximation of the Kullback-Leibler divergence ($D_{\mathrm{KL}}$) to estimate how much the posterior distribution differs from our gamma-distribution approximation. The largest deviation occurs for the worst-case outcome $m=1$ together with a small prior shape parameter, typically $k=3$. In this regime the posterior is strongly skewed, and the approximation error reaches a maximum of roughly $D_{\mathrm{KL}}\approx 0.09–0.10$ at intermediate ratios ${\tau }_{i+1}/\theta \gtrsim 1$. The accuracy improves systematically with increasing $k$. For example, at $k=10$ the peak divergence drops below 0.006, and for $k=20$ it falls below 0.001 across the entire range of ${\tau }_{i+1}/\theta$. See Supplemental Material [44] for further details on the validity of the approximation, also with respect to a normal distribution.

Because of the numerical precision of the controller, the actual ratio between ${\tau }_{i+1}$ and ${\widehat{T}}_{1,i}$ varies by approximately 10% in the vast majority of probe cycles.

Each probe cycle includes a $2.5\mathrm{\mu }\mathrm{s}$ readout period, followed by an approximately $8\mathrm{\mu }\mathrm{s}$ wait time to allow the resonator to cool down and fully deplete any residual photons from the readout pulse. Each Bayesian update takes $\approx 2.2\mathrm{\mu }\mathrm{s}$. The controller is programmed to start with an initial gamma distribution prior with $(k_0,{\theta }_0)=(3,450\mathrm{\mu }\mathrm{s})$. With this in mind, the adaptive waiting time in each cycle is set to ${\tau }_{i+1}\approx 0.51{\widehat{T}}_{1,i}$ (see Supplemental Material [44] for a motivation of the choice $c=0.51$), where ${\widehat{T}}_{1,i}$ is the estimator for $T_1$ after cycle $i$.

Here $(k_0,{\theta }_0)=(3,450\mathrm{\mu }\mathrm{s})$, $\alpha =0.11$, $\beta =0.14$. Considering also that the nonadaptive ${\tau }_{\mathrm{lin},i}$ is linearly stepped up to $\approx 600\mathrm{\mu }\mathrm{s}$, the average idle time from the point of view of the adaptive scheme is $t\approx 345\mathrm{\mu }\mathrm{s}$, mostly covered by the average $⟨{\tau }_{\mathrm{lin},i}⟩\approx 300\mathrm{\mu }\mathrm{s}$. The adaptive waiting time in each cycle is then set to ${\tau }_{i+1}\approx 0.98{\widehat{T}}_{1,i}$ (see Supplemental Material [44]).

$(k_0,{\theta }_0)=(3,600\mathrm{\mu }\mathrm{s})$, $\alpha =\beta =0.12$, and $c\approx 0.53$.