Error-probability relationship between cultivating T|+ and S|+

Establish whether, in magic state cultivation protocols, the logical error probability when cultivating the magic state T|+ is approximately twice that when cultivating the magic state S|+. Precisely quantify and justify this relationship in the setting relevant to physical-level distillation and clarify the conditions under which it holds.

Background

To enable efficient simulation with the stabilizer formalism, the paper replaces T gates with S gates, causing the distillation circuit to output S|+ instead of T|+. Interpreting the results for T|+ therefore depends on an assumed relationship between the logical error probabilities of cultivating T|+ versus S|+. The paper cites a conjectured factor-of-two relationship in logical error probability between the two cases and relies on it when comparing performance.

Validating or refuting this relationship is important for accurately mapping simulation outcomes with S|+ to expected outcomes with T|+ in cultivation-based physical-level distillation protocols, including the proposed lattice-surgery variant and the original magic state cultivation framework.