Universal fault-tolerant quantum computation with only transversal gates and error correction

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implementations of encoded unitary gates are highly desirable for
fault-tolerant quantum computation.  It is known, however, that
transversal gates alone cannot be computationally universal.  I will show
that the limitation on universality can be circumvented using only
fault-tolerant error correction, which is already required anyway.  This
result applies to ``triorthogonal'' stabilizer codes, which were recently
introduced by Bravyi and Haah for state distillation.  I will show that
triothogonal codes admit transversal implementation of the
controlled-controlled-Z gate, and then demonstrate a transversal Hadamard construction
which uses error correction to preserve the codespace. I will also discuss how
to adapt the distillation procedure of Bravyi and Haah to Toffoli gates,
improving on existing Toffoli distillation schemes.