T-vs-S error-rate doubling approximation in magic state cultivation

Establish whether, for all magic state cultivation circuits analyzed in the paper, the logical error rate of cultivating the T|+> magic state is accurately approximated by twice the logical error rate obtained when the same circuits replace each T gate with an S gate (i.e., cultivating S|+>), across the regimes considered (fault distances and uniform depolarizing circuit noise strengths).

Background

To make large-scale Monte Carlo simulations tractable, the authors use S-gate proxy circuits to estimate error rates for T-gate cultivation. Full state-vector simulations at fault distance 3 showed that cultivating T|+> has about twice the logical error rate of cultivating S|+>, motivating an assumption that doubling the S|+> error approximates the T|+> error. This approximation underpins the paper’s headline estimates and historical comparisons.

However, the approximation’s validity beyond the directly simulated cases (e.g., at higher target fault distance or different noise strengths) is uncertain and formal verification would strengthen the conclusions and calibrate the reliance on proxy circuits.