Full-Rank Constructions of Explicit Lossless Expanders and Good Small-Set qLTCs

We construct explicit two-sided lossless expanders whose binary adjacency matrices have full rank, and use them to obtain explicit asymptotically good quantum LDPC codes whose specified X- and Z-check matrices have full row rank and satisfy small-set soundness, or confinement, on a linear range. Our lossless expanders are a variant of those of Hsieh, Lubotzky, Mohanty, Reiner, and Zhang, with a free action of a finite 2-group. Under such an action, injectivity and surjectivity of the adjacency matrix are detected on a fixed-size matrix indexed by vertex orbits. Adding a sparse factor to the gadget at the middle vertices of a single type then enforces full rank, with a controlled loss in expansion. To obtain the 2-group action, we observe that reducing the generators of the Lubotzky-Phillips-Sarnak Ramanujan graphs modulo powers of 2, instead of modulo an odd prime, gives Ramanujan Cayley graphs on finite 2-groups. This construction may be of independent interest. Plugging two such expanders into the balanced product of Lin and Hsieh gives the quantum codes, since full rank of the input graphs is inherited by the check matrices. Our construction supplies the code family sought by Hong, Guo, and Lucas. Applying their analysis, within their framework and under their dynamical assumptions, gives quantum memories whose free-energy density is analytic at every positive temperature and whose guaranteed lifetime is $exp(Ω(N))$ at sufficiently low fixed positive temperature, where N is the blocklength. This improves the previously guaranteed stretched-exponential timescale.