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

François Arnault, Philippe Gaborit, Wouter Rozendaal, Nicolas Saussay, Gilles Zémor

TL;DR

The paper develops a refined Bravyi-Terhal bound for geometrically-local quantum codes defined on a $D$-dimensional lattice quotient $\mathbb{Z}^D/\Lambda$, bounding the minimum distance in terms of lattice geometry, locality $\rho$, and the Hermite constant $\gamma_D$. It then applies this bound to Abelian Two-Block Group Algebra (2BGA) codes, proving explicit distance upper bounds when the stabilizer weight is $w$ and the effective dimension is $D=w-2$. The approach blends a Gram-Schmidt basis optimization, parallelotope partitioning, and the Cleaning Lemma to translate geometric constraints into a nontrivial logical operator confined to a region, incurring a $\sqrt{\gamma_D}$ and a $\sqrt{D}$-dependent penalty. The results provide sharper, explicit distance bounds for 2BGA codes and connect lattice-theoretic constants to quantum code parameters, with implications for practical short-length quantum LDPC codes and their boundary conditions.

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 $\mathbb{Z}^D/Λ$ of $\mathbb{Z}^D$ of cardinality $n$, where $Λ$ 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 $ρ$, then the minimum distance $d$ of the code satisfies $d \leq m\sqrt{γ_D}(\sqrt{D} + 4ρ)n^\frac{D-1}{D}$ whenever $n^{1/D} \geq 8ρ\sqrt{γ_D}$, where $γ_D$ 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.

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

TL;DR

The paper develops a refined Bravyi-Terhal bound for geometrically-local quantum codes defined on a -dimensional lattice quotient , bounding the minimum distance in terms of lattice geometry, locality , and the Hermite constant . It then applies this bound to Abelian Two-Block Group Algebra (2BGA) codes, proving explicit distance upper bounds when the stabilizer weight is and the effective dimension is . The approach blends a Gram-Schmidt basis optimization, parallelotope partitioning, and the Cleaning Lemma to translate geometric constraints into a nontrivial logical operator confined to a region, incurring a and a -dependent penalty. The results provide sharper, explicit distance bounds for 2BGA codes and connect lattice-theoretic constants to quantum code parameters, with implications for practical short-length quantum LDPC codes and their boundary conditions.

Abstract

We present a modified version of the Bravyi-Terhal bound that applies to quantum codes defined by local parity-check constraints on a -dimensional lattice quotient. Specifically, we consider a quotient of of cardinality , where is some -dimensional sublattice of : we suppose that every vertex of this quotient indexes qubits of a stabilizer code , which therefore has length . We prove that if all stabilizer generators act on qubits whose indices lie within a ball of radius , then the minimum distance of the code satisfies whenever , where is the -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 with each submatrix representing an element of a group algebra over a finite abelian group.

Paper Structure

This paper contains 15 sections, 14 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Previous results
  4. Contributions
  5. High-level overview of Theorem \ref{['thm:BT bound']}
  6. Organisation of the paper
  7. Abelian Two-Block Group Algebra Codes
  8. Stabilizer and CSS codes
  9. Abelian 2BGA codes
  10. Bravyi-Terhal bound
  11. Lattices
  12. Hermite and Rankin constants
  13. Geometrically-local codes
  14. Proof of Theorem \ref{['thm:BT bound']}
  15. Minimum distance bound for Abelian 2BGA codes

Key Result

Proposition 3.2

Let $\Lambda \subset \mathbb{R}^D$ be a full rank lattice and $\| \cdot \|$ be any norm on $\mathbb{R}^D$. Then $\mathbb{R}^D/\Lambda$ is a metric space for the distance $dist$ induced by $\| \cdot \|$: Moreover, for any pair $(a,b) \in \mathbb{R}^D \times \mathbb{R}^D$ there exists $g_0 \in \Lambda$ such that $dist([a], [b]) = \| a - (b + g_0) \|$.

Theorems & Definitions (17)

  • Definition 3.1: Lattices
  • Proposition 3.2
  • Definition 3.3: Hermite constant H1850
  • Theorem 3.4: Minkowski
  • Definition 3.5: Rankin constant
  • Theorem 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Lemma 3.9
  • Lemma 3.10
  • ...and 7 more