Papers › Clifford gates with logical transversality for self-dual CSS codes
Clifford gates with logical transversality for self-dual CSS codes
Theerapat Tansuwannont, Yugo Takada, Keisuke Fujii
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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 ≥1 and prove necessary and sufficient conditions for the code to have a symplectic basis such that (1) transversal logical Hadamard gates ⊗ⱼ₌₁ᵏ H̅ⱼ can be implemented by transversal physical Hadamard gates ⊗ᵢ₌₁ⁿ Hᵢ, and (2) for any (a₁,…,aₖ)∈{-1,1}ᵏ, transversal logical phase gates ⊗ⱼ₌₁ᵏ S̅ⱼ^(aⱼ) can be implemented by transversal physical phase gates ⊗ᵢ₌₁ⁿ Sᵢ^(bᵢ) for some (b₁,…,bₙ)∈{-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.
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