A $\boldsymbol{2d \times d \times d}$ Spacetime Volume Implementation of a Logical S Gate in the Surface Code

Quantum Physics [Submitted on 15 Apr 2026] A \boldsymbol{2d \times d \times d} Spacetime Volume Implementation of a Logical S Gate in the Surface Code Yuga Hirai, Shota Ikari, Yosuke Ueno, Yasunari Suzuki The logical S gate implemented via twist defect braiding in the surface code is one of the major sources of overhead in fault-tolerant quantum computing, since an S-gate correction is required in every logical T-gate teleportation. Existing logical S-gate implementations require spacetime volumes of \(2d \times 2d \times d\) or \(2d \times 1.5d \times d\), where d is the code distance of the surface code. To the best of our knowledge, their circuit-level implementations have not yet been shown, hindering quantitative comparisons of fault distances and logical error rates. In this work, we provide these missing circuit-level implementations. Additionally, we propose a novel twist defect braiding protocol that reduces the spacetime volume to \(2d \times d \times d\). First, we construct an implementation of the proposed method using constant-length non-local gates, and then refine it to utilize only nearest-neighbor two-qubit gates on a square grid, without requiring additional two-qubit gate depth beyond that of standard syndrome extraction circuits. Through numerical simulations, we evaluate the fault distances and logical error rates for both existing and proposed methods. Our results show that, although the proposed method reduces the fault distance by one or three, its logical error rates remain comparable to those of existing methods at large code distances (\(d \ge 5\)) and at physical error rates near \(p = 10^{-3}\). This demonstrates that the proposed method is promising for near-term fault-tolerant quantum computing. Comments: 16 pages, 22 figures Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2604.13632 [quant-ph]   (or arXiv:2604.13632v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.13632 Focus to learn more Submission history From: Yuga Hirai [view email] [v1] Wed, 15 Apr 2026 09:00:52 UTC (4,599 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)