2025/02/07 by Arnault, François, Gaborit, Philippe, Rozendaal, Wouter +2 · 3 citations
#FOS: Computer and information sciences #FOS: Physical sciences #Information Theory (cs.IT) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2502.04995
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 ℤD/Λ of ℤD of cardinality n, where Λ is some D-dimensional sublattice of ℤD: 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 ρ, then the minimum distance d of the code satisfies d ≤ m√(γD)(√(D) + 4ρ)n^(D-1)/(D) whenever n1/D ≥ 8ρ√(γD), where γD 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 [A \vert B] with each submatrix representing an element of a group algebra over a finite abelian group.