A Variant of the Bravyi-Terhal Bound for Arbitrary Boundary Conditions

We present a modified version of the Bravyi-Terhal bound that applies to quantum codes defined by local parity-check constraints on a $D$-dimensional lattice quotient. Specifically, we consider a quotient $\mathbb{Z}^D/\Lambda$ of $\mathbb{Z}^D$ of cardinality $n$, where $\Lambda$ is some $D$-dimensional sublattice of $\mathbb{Z}^D$: we suppose that every vertex of this quotient indexes $m$ qubits of a stabilizer code $C$, which therefore has length $nm$. We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius $\rho$, then the minimum distance $d$ of the code satisfies $d \leq m\sqrt{\gamma_D}(\sqrt{D} + 4\rho)n^\frac{D-1}{D}$ whenever $n^{1/D} \geq 8\rho\sqrt{\gamma_D}$, where $\gamma_D$ is the $D$-dimensional Hermite constant. We apply this bound to derive an upper bound on the minimum distance of Abelian Two-Block Group Algebra (2BGA) codes whose parity-check matrices have the form $[\mathbf{A} \, \vert \, \mathbf{B}]$ with each submatrix representing an element of a group algebra over a finite abelian group.