Fault-tolerant recovery criteria for syndrome-mediated gates

Identify code and circuit properties that guarantee the outcome-conditioned Knill–Laflamme recovery condition for propagated fault paths in syndrome-mediated logical gates and allow the corresponding recovery to be implemented fault-tolerantly.

Background

The paper introduces an outcome-conditioned variant of the Knill–Laflamme criterion to determine whether fault paths that produce the same complete measurement record admit a common recovery. The protected 22-qubit and Golay constructions verify this condition through specialized concatenation, transported checks, and explicit recovery procedures, but they do not provide general criteria applicable to broader code and circuit families.

The unresolved problem is to characterize structural properties of stabilizer codes and gate circuits that ensure both recoverability of propagated faults and fault-tolerant implementation of the recovery itself. Such criteria could guide the design of selective-concatenation and transported-check constructions while accounting for protection throughout the complete logical gate rather than only in the initial code space.