Improved Rate-Distance Trade-Offs for Quantum Codes with Restricted Connectivity.

Abstract 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 $\dsl n,k,d \dsr$ quantum codes defined by local checks on the $D$-dimensional lattice must obey $k d^{2/(D-1)} \le O(n)$. 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.