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