arXiv subjects

Nicolas Saussay

Publications and source records attributed to Nicolas Saussay.

4 recordsLinked to original sources

arXiv2025

(2,2)-GB Codes: Classification and Comparison with weight-4 Surface Codes

Generalized Bicycle (GB) codes offer a compelling alternative to surface codes for quantum error correction. This paper focuses on (2,2)-Generalized Bicycle codes, constructed from pairs of binary circulant matrices with two non-zero elements per row. Leveraging a lower bound on their minimum distance, we construct three novel infinite families of optimal (2,2)-GB codes with parameters [[ 2n^2, 2, n ]], [[ 4r^2, 2, 2r ]], and [[(2t + 1)^2 + 1, 2, 2t + 1 ]]. These families match the performance of Kitaev's toric code and the best 2D weight-4 surface codes, reaching known theoretical limits. In particular, the second family breaks a long-held belief by providing optimal even-distance GB codes, previously deemed impossible. All are CSS codes derived from Cayley graphs. Recognizing that standard equivalence relations do not preserve their CSS structure, we introduce a CSS-preserving equivalence relation for rigorous comparison of Cayley graph-based CSS codes. Under this framework, the first two families are inequivalent to all previously known optimal weight-4 2D surface codes, while the third family is equivalent to the best-known odd-distance 2D surface code. Finally, we classify all extremal, non-equivalent (2,2)-GB codes with length below 200 and present a comparison table with existing notable 2D weight-4 surface codes.

cs.IT↗

arXiv2025

On the Generalization of Kitaev Codes as Generalized Bicycle Codes

Surface codes have historically been the dominant choice for quantum error correction due to their superior error threshold performance. However, recently, a new class of Generalized Bicycle (GB) codes, constructed from binary circulant matrices with three non-zero elements per row, achieved comparable performance with fewer physical qubits and higher encoding efficiency. In this article, we focus on a subclass of GB codes, which are constructed from pairs of binary circulant matrices with two non-zero elements per row. We introduce a family of codes that generalizes both standard and optimized Kitaev codes for which we have a lower bound on their minimum distance, ensuring performance better than standard Kitaev codes. These codes exhibit parameters of the form $ [| 2n , 2, \geq \sqrt{n} |] $ where $ n$ is a factor of $ 1 + d^2 $. For code lengths below 200, our analysis yields $21$ codes, including $7$ codes from Pryadko and Wang's database, and unveils $14$ new codes with enhanced minimum distance compared to standard Kitaev codes. Among these, $3$ surpass all previously known weight-4 GB codes for distances $4$, $8$, and $12$.

cs.IT↗

arXiv2025

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 $\mathbb{Z}^D/Λ$ 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.

quant-ph↗

arXiv2025

Upper Bounds on the Minimum Distance of Structured LDPC Codes

We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form $[\mathbf{C} \vert \mathbf{M}]$ where $\mathbf{C}$ is circulant and of column weight $2$, and $\mathbf{M}$ has fixed column weight $r \geq 3$ and row weight at least $1$. These codes are of interest because they are LDPC codes which come with a natural linear-time encoding algorithm. We show that the minimum distance of these codes is in $O(n^{\frac{r-2}{r-1} + ε})$, where $n$ is the code length and $ε> 0$ is arbitrarily small. This improves the previously known upper bound in $O(n^{\frac{r-1}{r}})$ on the minimum distance of such codes.

cs.IT↗