The disjointness of stabilizer codes and limitations on fault-tolerant logical gates

Understand

Stabilizer codes are a simple and successful class of quantum error-correcting codes.

• Yet this success comes in spite of some harsh limitations on the ability of these codes to fault-tolerantly compute.
• Here we introduce a new metric for these codes, the disjointness, which, roughly speaking, is the number of mostly non-overlapping representatives of any given non-trivial logical Pauli operator.
• We use the disjointness to prove that transversal gates on error-detecting stabilizer codes are necessarily in a finite level of the Clifford hierarchy.