Reed-Muller and Transversal T
The 15-qubit Reed-Muller code's transversal T gate makes it a natural core for magic-state distillation and code-switching schemes.
Why transversal T is special
The T gate, a pi/4 phase rotation, is the standard non-Clifford gate that lifts the Clifford group to universality. Applying it fault tolerantly is the hardest part of building a universal quantum computer, because in the surface code it must be teleported in through a distilled magic state. A code where T is simply applied qubit by qubit removes one layer of that machinery.
The 15-qubit code
The [[15,1,3]] quantum Reed-Muller code has exactly this property. Applying T-dagger to each of the 15 physical qubits implements a logical T. The code is a CSS code whose X and Z checks come from nested Reed-Muller codes, and its structure ensures the physical T operations combine into a single clean logical T with no leftover Clifford correction beyond a known Pauli frame update.
- Transversal gates cannot spread a single error into an uncorrectable one, the core of fault tolerance.
- The 15-qubit code trades transversal Clifford gates away to gain transversal T.
- Its distance is only three, so it corrects single errors, adequate inside distillation.
- Pairing it with the Steane code covers the full Clifford-plus-T set.
This code sits at the heart of the classic 15-to-1 magic-state distillation routine, which takes fifteen noisy T states and outputs one much cleaner T state by using the code's structure to detect the most likely errors. It also enables code switching, where information moves between a code with transversal Clifford gates and this code to apply T.
The lesson generalizes: Eastin-Knill forbids any one code from having a transversal universal set, so real machines assemble universality from complementary codes and distillation, and the Reed-Muller code is a workhorse for the T portion.