Fig. 6: Ancilla-qubit leakage detection.

a–d Precision-recall curves for the ancilla-qubit HMMs over 4 × 10⁴ runs of 50 QEC cycles each. In a, b the HMMs rely only on the observed defects on the neighboring stabilizers. In c–f the HMMs further get the in-phase component I_m of the analog readout as input, from which \({p}_{m}^{{\mathcal{L}}}\) is extracted. The dotted line corresponds to a random guess classifier for which \({\mathcal{P}}\) is equal to the fraction of leakage events over all QEC cycles and runs. As ancilla-qubit leakage is directly measured, \({{\mathcal{P}}}_{{\rm{DM}}}=1\) for all values of \({\mathcal{R}}\) (not shown). Insets in c, d: the HMM optimality \({\mathcal{O}}\) as a function of the discrimination fidelity \({F}^{{\mathcal{L}}}\) between \(\left|1\right\rangle\) and \(\left|2\right\rangle\). The corresponding error bars (extracted over 2 × 10⁴ runs of 20 QEC cycles each) are smaller than the symbol size. The vertical dashed line corresponds to the experimentally measured \({F}^{{\mathcal{L}}}=88.4 \%\). e, f Average response in time of the ancilla-qubit HMMs (diamonds) to leakage, in comparison to the actual leakage probability extracted directly from the readout (dashed), extracted over 4 × 10⁴ runs of 50 QEC cycles each. The average is computed by selecting single realizations where the qubit was in the computational subspace for at least 3 QEC cycles and then in the leakage subspace for 5 or more.