Transversal Clifford and T-gate codes of short length and high distance

Abstract: The non-local interactions in several quantum device architectures 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. To this aim, we construct three kinds of codes encoding a single logical qubit for distances up to $31$. First, we construct the smallest known doubly even codes, all of which admit a transversal implementation of the Clifford group. Applying a doubling procedure [arXiv:1509.03239] to such codes yields the smallest known weak triply even codes for the same distances and number of encoded qubits. This second family of codes admit a transversal implementation of the logical $\texttt{T}$-gate. Relaxing the triply even property, we obtain our third family of triorthogonal codes with an even lower overhead at the cost of requiring additional Clifford gates to achieve the same logical operation. To our knowledge, these are the smallest known triorthogonal codes for their respective distances. While not qLDPC, the stabilizer generator weights of the code families with transversal $\texttt{T}$-gates scale roughly as the square root of their lengths.