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.