We present a modified version of the Bravyi-Terhal bound that applies to quantum codes defined by local parity-check constraints on aD-dimensional lattice quotient. Specifically, we consider a quotient ZD/Λ of ZDof cardinality ℓ, where Λ is someD-dimensional sublattice of ZD: we suppose that every vertex of this quotient indexesmqubits of a stabilizer codeC, which therefore has lengthn=mℓ. We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius ρ, then the minimum distancedof the code satisfiesd≤m√ γD( √D+ 4ρ)ℓD−1/D, where γDis theD-dimensional Hermite constant. We then 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 [A|B] with each submatrix representing an element of a group algebra over a finite abelian group.

Kitaev’s toric code is arguably the most studied quantum code and is expected to be implemented in future generations of quantum computers. The renormalisation decoders introduced by Duclos-Cianci and Poulin exhibit one of the best trade-offs between accuracy and efficiency, with a time complexity inO(nlog2n). One question that was left open is how they handle worst-case or adversarial errors, i.e. what is the order of magnitude of the smallest weight of an error pattern that will be wrongly decoded. We initiate such a study involving a simple hard-decision and deterministic version of a renormalisation decoder. We exhibit an uncorrectable error pattern whose weight scales liked1/2and prove that the decoder corrects all error patterns of weight less than 5/6dlog2(6/5), wheredis the minimum distance of the toric code.

Kitaev's toric code is arguably the most studied quantum code and is expected to be implemented in future generations of quantum computers. The renormalisation decoders introduced by Duclos-Cianci and Poulin exhibit one of the best trade-offs between efficiency and speed, but one question that was left open is how they handle worst-case or adversarial errors, i.e. what is the order of magnitude of the smallest weight of an error pattern that will be wrongly decoded. We initiate such a study involving a simple hard-decision and deterministic version of a renormalisation decoder. We exhibit an uncorrectable error pattern whose weight scales like d1/2and prove that the decoder corrects all error patterns of weight less than $\frac{5}{6}{d^{{{\log }_2}(6/5)}}$, where d is the minimum distance of the toric code.