Clifford gates with logical transversality for self-dual CSS codes

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 $k\ge 1$ and prove necessary and sufficient conditions for the code to have a symplectic basis such that (1) transversal logical Hadamard gates ${⨂}_{j=1}^k{\bar{H}}_j$ can be implemented by transversal physical Hadamard gates ${⨂}_{i=1}^n{H}_i$, and (2) for any $(a_1,\dots ,a_k)\in \{-1,1{\}}^k$, transversal logical phase gates ${⨂}_{j=1}^k{\bar{S}}_j^{a_j}$ can be implemented by transversal physical phase gates ${⨂}_{i=1}^n{S}_i^{b_i}$ 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.