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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.19790 |
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| _version_ | 1866909552082944000 |
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| author | Tansuwannont, Theerapat Takada, Yugo Fujii, Keisuke |
| author_facet | Tansuwannont, Theerapat Takada, Yugo Fujii, Keisuke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19790 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Clifford gates with logical transversality for self-dual CSS codes Tansuwannont, Theerapat Takada, Yugo Fujii, Keisuke Quantum Physics 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. |
| title | Clifford gates with logical transversality for self-dual CSS codes |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2503.19790 |