Extended Data Fig. 8. State-preparation correction for Bell state population.

We draw the probability tree, starting from circuit initialization (state preparation), and track the evolution up to the measurement stage. From this, we can correct for state preparation errors. Following erasure-cooling as part of the state preparation, we have the ideal initial state and erroneous initial states with respective probabilities. We track their evolution under the Bell state generation circuit and see how each initial state contributes to the population distribution before the measurement. The SPAM corrected values, labelled by $P_b^c$, where b labels the different two-qubit bitstring, are the quantities that we want to evaluate. The pathway for $\mid \mathrm{lost},1⟩$ and $\mid 01⟩$ can be inferred easily from the clear symmetry with $\mid 1,\mathrm{lost}⟩$ and $\mid 10⟩$. For simplicity, $\mid 00⟩,\mid \mathrm{lost},\mathrm{lost}⟩,\mid 0,\mathrm{lost}⟩,\mid \mathrm{lost},0⟩$ are not shown due to their small probability on the order of ${\epsilon }_d^2,{\epsilon }_l^2\approx 10^{-5}$.