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

TL;DR

This paper improves understanding of how limited qubit connectivity affects quantum code parameters, providing tighter bounds on code dimension and distance based on the connectivity graph's properties.

Contribution

It extends previous trade-off bounds to a broader class of quantum codes, establishing tighter relations between code parameters and graph separator sizes.

Findings

Tighter dimension-distance trade-off as a function of separator size.

Distance bounds applicable to all stabilizer codes with certain separation profiles.

Generalization beyond geometrically-local codes.

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 (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.