Transversality Across Code Families
Which gates a code can apply transversally depends on its structure, and comparing families shows why no single code gives a universal transversal set.
Transversality is code-specific
A transversal gate applies an independent physical gate to each qubit of a code block, so a single fault stays confined to one qubit and cannot spread into an uncorrectable error. Which logical gates a given code realizes this way is fixed by its algebraic structure, and it varies sharply from one code family to another.
A comparison
CSS codes automatically have transversal CNOT between two identical blocks, and often transversal Hadamard and phase gates. The Steane code gives the full transversal Clifford group. The 15-qubit Reed-Muller code gives transversal T but not the full Clifford group. Two-dimensional color codes give all Clifford gates transversally, while the surface code gives essentially none beyond Pauli operations.
- CSS structure guarantees transversal CNOT between blocks.
- Steane [[7,1,3]]: transversal Clifford, no transversal T.
- Reed-Muller [[15,1,3]]: transversal T, partial Clifford.
- 2D color codes: full transversal Clifford; surface code: almost none.
The pattern is a consequence of the Eastin-Knill theorem: a code with a transversal universal gate set cannot detect errors, so every useful code is missing at least one gate transversally. The Steane and Reed-Muller codes are complementary, each supplying what the other lacks, which is why code switching between them can reach universality.
This complementarity guides architecture. A machine can pick a code with cheap transversal Cliffords and obtain the T gate by switching codes, distilling magic states, or gauge fixing. The choice of code family is therefore inseparable from the choice of how to implement the non-Clifford gate.