Fault-tolerant operations under quantum code extension

Extend a quantum error-correcting code to successively greater distances while preserving fault-tolerant implementations of all logical operations in the extended code.

Background

The SAT formulation guarantees error detectability but does not impose conditions ensuring that logical operations have fault-tolerant implementations. The authors observe that extending a code can therefore compromise the fault tolerance of operations that were available for the original code. Developing extensions that preserve fault-tolerant operations, potentially through sequences of codes with increasing distance and transversal operations, is explicitly identified as future work.