Good quantum codes enable new transversal controlled Z logical gates
Transversal non-Clifford gates on good quantum locally testable codes
Information Theory
Summary
Quantum computing requires ways to protect information from errors using quantum error-correcting codes. The authors show how to create a family of such codes that not only protect well but also support a special kind of logical operation called transversal multi-controlled-Z gates. These operations can be performed quickly and in parallel across the encoded data, which is valuable for building practical quantum computers. Their constructions use advanced mathematical tools involving geometric spaces and algebra, improving on previous approaches.
What this means in practice
- •For quantum hardware engineers: Implement more efficient fault-tolerant logical gates on quantum error-correcting architectures using codes that allow transversal multi-controlled-Z operations.
- •For quantum software developers: Design quantum algorithms that leverage transversal multi-controlled-Z gates to reduce gate depth and error accumulation in logical operations.
A theory result. No direct application yet.
Authors
Yiming Li, Zimu Li, Zi-Wen Liu
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.