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

2025-02-07

AI Summary

This work improves the classical Bravyi–Terhal distance bound for local stabilizer codes defined on quotient spaces of D-dimensional lattices, extending its applicability to arbitrary boundary conditions—including both periodic and non-periodic topologies. Methodologically, we generalize the bound to general D-dimensional lattice quotients Λ and incorporate the Hermite constant γ_D to explicitly characterize the interdependence of code distance d on dimension D, lattice radius ρ, and code length n. Our main contributions are: (i) a tight upper bound on the distance, d ≤ m√γ_D(√D + 4ρ)n^(D−1)/D; and (ii) the first universal distance upper bound for Abelian two-block group algebra codes, enabling unified distance analysis of local stabilizer codes across diverse topological settings. The result bridges geometric coding theory and lattice-based quantum error correction, providing a dimensionally explicit and topology-agnostic framework for bounding code distance.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

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 $ ho$, then the minimum distance $d$ of the code satisfies $d leq msqrt{gamma_D}(sqrt{D} + 4 ho)n^frac{D-1}{D}$ whenever $n^{1/D} geq 8 hosqrt{gamma_D}$, 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.

Problem

• Extended Bravyi-Terhal bound
• Quantum codes on lattices
• Minimum distance upper bound

Innovation

• Modified Bravyi-Terhal bound
• Quantum codes on lattice quotient
• Upper bound for 2BGA codes