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

Yiming Li, Zimu Li, Zi-Wen Liu

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 $\tilde{\Theta }(1)$, comb…