Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction

TLDR

Transversal encoded gates are highly desirable for fault‑tolerant quantum computation, but alone they cannot provide universality; they must be combined with specially distilled resource states to achieve universal operations. The authors demonstrate that triorthogonal stabilizer codes enable a transversal implementation of the controlled‑controlled‑Z (Toffoli) gate. They build a universal fault‑tolerant gate set using only transversal Toffoli, transversal Hadamard, and fault‑tolerant error correction, and they modify the Bravyi–Haah distillation protocol to produce Toffoli states more efficiently. This construction yields a universal set without requiring state distillation and improves existing Toffoli distillation schemes.

Abstract

Transversal implementations of encoded unitary gates are highly desirable for fault-tolerant quantum computation. Though transversal gates alone cannot be computationally universal, they can be combined with specially distilled resource states in order to achieve universality. We show that "triorthogonal" stabilizer codes, introduced for state distillation by Bravyi and Haah [Phys. Rev. A 86, 052329 (2012)], admit transversal implementation of the controlled-controlled-Z gate. We then construct a universal set of fault-tolerant gates without state distillation by using only transversal controlled-controlled-Z, transversal Hadamard, and fault-tolerant error correction. We also adapt the distillation procedure of Bravyi and Haah to Toffoli gates, improving on existing Toffoli distillation schemes.