Transversal non-Clifford gates on good quantum locally testable codes

2026-09-22

AI Summary

通过在良好量子局部可测试码上应用特定门框架和覆盖空间方法，实现具有渐近最优参数的非平凡横向逻辑多控制-Z门。

Abstract

We achieve nontrivial transversal logical multi-controlled-$Z$ gates with asymptotically optimal parameters simultaneously on quantum low-density parity check codes and quantum locally testable codes, by applying the gate framework of [arXiv:2604.01874] to the recent good qLTC construction of [arXiv:2609.20780]. To this end, we use the covering space method to construct a nonzero cup product pairing on a finite arithmetic cubical complex. This differs from the previous construction of almost-good codes, whose base space is a hypergraph product. We express the pairing as a coefficient in a product of Moore determinants and prove polynomial nonvanishing by a bipartite multigraph specialization. We then construct covering spaces to obtain an asymptotic family of good qLTCs on which the pulled-back pairing induces the desired nontrivial transversal logical action.

Problem

• transversal non-Clifford gates
• quantum locally testable codes
• multi-controlled-Z gates

Innovation

• transversal logical multi-controlled-Z gates
• quantum locally testable codes (qLTCs)
• covering space method
• cup product pairing
• asymptotically optimal parameters