  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 (BT) and Bravyi, Poulin & Terhal
(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 $k d^{2/(D-1)} \le O(n)$. Baspin and Krishna
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.
