Error-correcting codes for fermionic quantum simulation

We provide ways to simulate fermions by qubits on 2d lattices using $mathbb{Z}_2$ gauge theories (stabilizer codes). By studying the symplectic automorphisms of the Pauli module over the Laurent polynomial ring, we develop a systematic way to increase the code distances of stabilizer codes. We identify a family of stabilizer codes that can be used to simulate fermions with code distances of $d=2,3,4,5,6,7$ such that any $lfloor frac{d-1}{2} floor$-qubit error can be corrected. In particular, we demonstrate three stabilizer codes with code distances of $d=3$, $d=4$, and $d=5$, respectively, with all stabilizers and logical operators shown explicitly. The syndromes for all Pauli errors are provided. Finally, we introduce a syndrome-matching method to compute code distances numerically.