We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces -- composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock-state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group $SU(q)$, we prove that many codes and their gates in $SU(q)$ can be inter-converted between the three state spaces. We construct new code instances for all three spaces using classical $\ell_1$ codes and Tverberg's theorem, a classic result from convex geometry. We obtain new families of quantum codes with distance that scales almost linearly with the code length $N$ by constructing $\ell_1$ codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This improves upon the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, new bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.

翻译：我们建立了一个框架，用于为三类空间——多量子比特或量子比特复合置换不变空间、多玻色模复合恒定激发福克态空间，以及原子、离子和分子的单体核态空间——构造量子纠错码与逻辑门。通过将这三类空间与离散单纯形及李群$SU(q)$的表示相联系，我们证明了$SU(q)$中的许多码及其门操作可以在三种态空间之间相互转换。我们利用经典$\ell_1$码与凸几何中的经典结果——Tverberg定理，为所有三类空间构造了新的码实例。通过基于称为Sidon集的组合模式构造$\ell_1$码，并利用其Tverberg划分，我们获得了距离随码长$N$几乎线性增长的新量子码族。这改进了现有所有态空间的设计方案。我们给出了具有比已知同类参数码更短长度或更低总自旋/激发的码的显式构造、具有奇异高斯门的新型玻色码，以及距离大于已知构造的短码示例。