Clifford gates with logical transversality for self-dual CSS codes (2503.19790v1)

Abstract: Quantum error-correcting codes with high encoding rate are good candidates for large-scale quantum computers as they use physical qubits more efficiently than codes of the same distance that encode only a few logical qubits. Some logical gate of a high-rate code can be fault-tolerantly implemented using transversal physical gates, but its logical operation may depend on the choice of a symplectic basis that defines logical Pauli operators of the code. In this work, we focus on $[![n,k,d]!]$ self-dual Calderbank-Shor-Steane (CSS) codes with $k \geq 1$ and prove necessary and sufficient conditions for the code to have a symplectic basis such that (1) transversal logical Hadamard gates $\bigotimes_{j=1}^{k} \bar{H}j$ can be implemented by transversal physical Hadamard gates $\bigotimes{i=1}^{n} H_i$, and (2) for any $(a_1,\dots,a_k)\in{-1,1}^k$, transversal logical phase gates $\bigotimes_{j=1}^{k} \bar{S}j^{a_j}$ can be implemented by transversal physical phase gates $\bigotimes{i=1}^{n} S_i^{b_i}$ for some $(b_1,\dots,b_n)\in{-1,1}^n$. Self-dual CSS codes satisfying the conditions include any codes with odd $n$. We also generalize the idea to concatenated self-dual CSS codes and show that certain logical Clifford gates have multiple transversal implementations, each by logical gates at a different level of concatenation. Several applications of our results for fault-tolerant quantum computation with low overhead are also provided.