Improved rate-distance trade-offs for quantum codes with restricted connectivity

For quantum error-correcting codes to be realizable, it is important that the qubits subject to the code constraints exhibit some form of limited connectivity. The works of Bravyi & Terhal (NJP 2009) (BT) and Bravyi, Poulin & Terhal (PRL 2010) (BPT) established that geometric locality constrains code properties---for instance n,k,d quantum codes defined by local checks on the D-dimensional lattice must obey k d2/(D-1) ≤ O(n). Baspin and Krishna (Quantum 2022) studied the more general question of how the connectivity graph associated with a quantum code constrains the code parameters. These trade-offs apply to a richer class of codes compared to the BPT and BT bounds, which only capture geometrically-local codes. We extend and improve this work, establishing a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph. We also obtain a distance bound that covers all stabilizer codes with a particular separation profile, rather than only LDPC codes.