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

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.