Quantum Reed-Muller Codes
Built from classical Reed-Muller codes, this family yields CSS codes with useful transversal non-Clifford gates, at the cost of a low encoding rate.
From classical to quantum
Classical Reed-Muller codes RM(r,m) are built from low-degree Boolean polynomials sampled over all bit strings. They form a nested family in which RM(r,m) contains RM(r-1,m), and this nesting is exactly what the CSS construction needs: one code supplies the X-checks, its subcode supplies the Z-checks. The result is a family of quantum CSS codes indexed by r and m.
Transversal non-Clifford gates
The reason these codes are studied is their gate structure. The 15-qubit quantum Reed-Muller code encodes one logical qubit with distance three and admits a transversal T gate, applied as a T or T-dagger on each physical qubit. Transversal T is forbidden in the surface code and most CSS codes, so this property is rare and valuable.
- The construction relies on the self-orthogonality guaranteed by Reed-Muller nesting.
- The 15-qubit code has parameters [[15,1,3]] and transversal T.
- Its dual-like partner, the Steane code, has transversal Clifford gates instead.
- Together such codes support code switching to reach a universal set.
No single stabilizer code can have a transversal universal gate set, a fact fixed by the Eastin-Knill theorem. Quantum Reed-Muller codes sidestep this by pairing with a complementary code: one supplies transversal Clifford gates, the other supplies transversal T, and gauge fixing or code switching moves between them.
The drawback is rate and distance. To get transversal T at larger distance, the codes grow quickly in qubit count, and their thresholds are modest. They are therefore used as building blocks, especially inside magic-state distillation circuits, rather than as the top-level memory of a large machine.