Symmetric Self-Dual Quantum Codes on High Dimensional Expanders
This addresses the problem of building fault-tolerant quantum codes with efficient logical operations for quantum computing, representing a novel advancement beyond prior incremental product-based constructions.
The authors constructed a family of constant-rate self-dual quantum LDPC codes on high-dimensional expanders, achieving the first such codes with rich symmetry and an extensive set of logical generators for fault-tolerant gates.
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.