Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes
This enables more efficient quantum computation by combining high-performance codes with essential non-Clifford gates, advancing fault-tolerant quantum computing.
The paper tackles the problem of implementing fault-tolerant non-Clifford gates on quantum error-correcting codes, achieving transversal logical multi-controlled-Z gates on quantum LDPC and locally testable codes with nearly optimal parameters such as [[N, Θ(N), ̃Θ(N)]] and soundness ̃Θ(1).
We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,Î(N),\tildeÎ(N)]\!]$ quantum low-density parity-check codes and $[\![N,Î(N),\tildeÎ(N)]\!]$ quantum locally testable codes with soundness $\tildeÎ(1)$, 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.