We construct a family of constant-rate highly-symmetric self-dual qLDPC codes on high dimensional expanders. This is the first self-dual code constructed on high dimensional expanders and also the first such code with a rich (e.g. transitive) symmetry group, whose order exceeds the number of qubits. From this symmetry, we identify an extensive set of logical generators that act as permutations or diagonal gates, as well as a handful of other interesting gates (including the logical swap-Hadamard from self-duality). These advantages over prior constructions are in large part due to the fact that our codes are the first to be explicitly defined on expanding (non-product) simplicial complexes. Indeed, our work develops a broader framework toward utilizing high dimensional expanders to construct highly performant quantum codes with fault tolerant gates. While asymptotically good qLDPC codes have been constructed on 2D HDX built from products of graphs, these product constructions have a number of limitations, such as a lack of structure useful for fault-tolerant logic. Our framework for (fold-)transversal logical gates naturally utilizes symmetric non-product simplicial high dimensional expanders, and we demonstrate concretely through our 2D code family how this framework gives a rich set of fault-tolerant logical generators.

翻译：我们在高维扩展器上构建了一族恒定速率、高度对称的自对偶qLDPC码。这是在高维扩展器上构建的首个自对偶码，也是首个具有丰富（例如传递）对称群且其阶数超过量子比特数量的此类编码。基于该对称性，我们识别出一组广泛的逻辑生成元，它们表现为置换门或对角门，以及少量其他有趣的门（包括由自对偶性产生的逻辑交换-哈达玛门）。相较于先前构造，这些优势在很大程度上源于我们的编码是首个明确定义在扩展（非乘积）单纯复形上的编码。事实上，我们的工作开发了一个更广泛的框架，旨在利用高维扩展器构建具有容错门的高性能量子码。虽然渐近良好的qLDPC码已在基于图乘积构建的二维高维扩展器上实现，但这些乘积构造存在诸多限制，例如缺乏对容错逻辑有用的结构。我们关于（折叠）横向逻辑门的框架自然地利用了对称的非乘积单纯高维扩展器，并通过我们的二维编码族具体展示了该框架如何提供一组丰富的容错逻辑生成元。