Subsystem codes with spatially local generators
Sergey Bravyi · Physical Review A · 2011
We study subsystem codes whose gauge group has local generators in two-dimensional (2D) geometry. It is shown that there exists a family of such codes defined on lattices of size $L\ifmmode\times\else\texttimes\fi{}L$ with the number of logical qubits $k$ and the minimum distance $d$ both proportional to $L$. The gauge group of these codes involves only two-qubit generators of type $\mathit{XX}$ and $\mathit{ZZ}$ coupling nearest-neighbor qubits (and some auxiliary one-qubit generators). Our proof is not constructive as it relies on a certain version of the Gilbert-Varshamov bound for classical codes. Along the way, we introduce and study properties of generalized Bacon-Shor codes that might be of independent interest. Secondly, we prove that any 2D subsystem $[n,k,d]$ code with spatially local generators obeys upper bounds $\mathit{kd}=O(n)$ and ${d}^{2}=O(n)$. The analogous upper bound proved recently for 2D stabilizer codes is ${\mathit{kd}}^{2}=O(n)$. Our results thus demonstrate that subsystem codes can be more powerful than stabilizer codes under the spatial locality constraint.