Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908933430444032 |
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| author | Li, Yiming Li, Zimu Liu, Zi-Wen |
| author_facet | Li, Yiming Li, Zimu Liu, Zi-Wen |
| contents | We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum low-density parity-check codes and $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum locally testable codes with soundness $\tildeΘ(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates for the first time. Remarkably, our proofs are almost entirely algebraic-topological, showing that such presumably intricate logical gates naturally arise as a fundamental topological phenomenon. We develop a general framework for constructing a rich new family of homological invariant forms which we call ''cupcap gates'' that induce transversal logical multi-controlled-$Z$ and, building on insights from [Li et al., arXiv:2603.25831], covering space methods to certify their nontriviality. The claimed almost-good code results follow immediately as examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01874 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes Li, Yiming Li, Zimu Liu, Zi-Wen Quantum Physics Information Theory Mathematical Physics We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum low-density parity-check codes and $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum locally testable codes with soundness $\tildeΘ(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates for the first time. Remarkably, our proofs are almost entirely algebraic-topological, showing that such presumably intricate logical gates naturally arise as a fundamental topological phenomenon. We develop a general framework for constructing a rich new family of homological invariant forms which we call ''cupcap gates'' that induce transversal logical multi-controlled-$Z$ and, building on insights from [Li et al., arXiv:2603.25831], covering space methods to certify their nontriviality. The claimed almost-good code results follow immediately as examples. |
| title | Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes |
| topic | Quantum Physics Information Theory Mathematical Physics |
| url | https://arxiv.org/abs/2604.01874 |