New local stabilizer codes from local classical codes

Amongst quantum error-correcting codes the surface code has remained of particular promise as it has local and very low-weight checks, even despite only encoding a single logical qubit no matter the lattice size. In this work we discuss new local and low-weight stabilizer codes which are obtained from the recent progress in $2D$ local classical codes. Of note, we construct codes with weight and qubit use count of $5$ while being able to protect the information with high distance, or greater logical count. We also consider the Fibonacci code family which generates weight and qubit use count of $6$ while having parameters $[[O(l^3),O(l),\mathrm{\Omega }(l)]]$. While other weight-reduction methods centered on lowering the weight without regard to locality, this work achieves very low-weight and geometric locality. This work is exhaustive over translated classical generators of size $3\times 3$ and up to size $17\times 17$ classical bit grids.