Quantum group codes for non-Clifford logic: enhanced decoding, addressability and parallelizability

Abstract:We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-$Z$ gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-$Z$ gate parallelizability. More precisely, on input a good quantum AG code over the alphabet $\mathbb F_q$ with transversal $\mathsf{C}^m\mathsf Z$ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet $\mathbb F_{q^2}$ supporting a transversal $\mathsf{C}^m\mathsf Z$ gate as well as addressable and parallelizable $\mathsf{C}^{m-1}\mathsf Z$ gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2606.27211 [quant-ph]

(or arXiv:2606.27211v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.27211 Focus to learn more arXiv-issued DOI via DataCite

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