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

Abstract

We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[[N,\Theta (N),\tilde{\Theta }(N)]]$ quantum low-density parity-check (qLDPC) codes with soundness $\tilde{\Theta }(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates on qLDPC and quantum locally testable codes for the first time. Remarkably, our proofs proceed through highly general algebraic arguments. Building on insights from [Li et al.,~arXiv:2603.25831], we develop a general covering space framework for constructing and computing a rich family of cohomological invariant forms on sheaf codes that induce transversal logical multi-controlled-$Z$. To certify their nontriviality, we further demonstrate the existence of two-way product-expanding punctured Reed--Solomon codes, which is striking in light of the many negative examples for the product expansion behavior of ordinary Reed--Solomon codes. This approach directly overcomes the previous obstruction to realizing nontrivial logical operations while simultaneously preserving the code parameters. The claimed almost-good code results follow immediately as examples.

arXiv commentIn v1, the cap product used there did not satisfy the required Leibniz rule, hence the cupcap gate argument is invalid. We now fix the problem and retain the results by a substantially modified argument relying on new product-expanding local codes