2025/03/25 by Tansuwannont, Theerapat, Takada, Yugo, Fujii, Keisuke · 2 citations
#FOS: Physical sciences #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2503.19790
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 \bigotimesj=1k Hj can be implemented by transversal physical Hadamard gates \bigotimesi=1n Hi, and (2) for any (a1,…,ak)∈\-1,1\k, transversal logical phase gates \bigotimesj=1k Sjaj can be implemented by transversal physical phase gates \bigotimesi=1n Sibi 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.