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arXiv · 2609.26691

Transversal non-Clifford gates on good quantum locally testable codes

Yiming Li·Zimu Li·Zi-Wen Liu
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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.

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Yiming Li, Zimu Li, Zi-Wen Liu. 2026-09-22. Transversal non-Clifford gates on good quantum locally testable codes. https://arxiv.org/abs/2609.26691

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