Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes

• URL: http://arxiv.org/abs/2604.01874v1
• Date: Thu, 02 Apr 2026 10:30:00 GMT
• Title: Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes
• Abstract summary: We show nontrivial logical multi-controlled-$Z$ gates on $[![N,(N),tilde(N)]!]$ quantum low-density parity-check codes and $[![N,(N),tilde(N)]!]$ quantum locally testable codes with soundness.
• Score: 12.13900619434341
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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