High-distance codes with transversal Clifford and T-gates

The non-local interactions in several quantum devices allow for the realization of more compact quantum encodings while retaining the same degree of protection against noise. Anticipating that short to medium-length codes will soon be realizable, it is important to construct stabilizer codes that, for a given code distance, admit fault-tolerant implementations of logical gates with the fewest number of physical qubits. We extract high-distance doubly even codes from the quantum quadratic-residue code family that admit a transversal implementation of the single-qubit Clifford group. Applying a doubling procedure [arXiv:1509.03239] to such codes yields a family of high-distance triply even codes which admit a transversal implementation of the logical T-gate. To our knowledge, both code families require a lower qubit overhead than any other qubit stabilizer code of the same distance which can realize their respective gates.