arXiv subjects

Wouter Rozendaal

Publications and source records attributed to Wouter Rozendaal.

4 recordsLinked to original sources

arXiv2025

Small quantum Tanner codes from left--right Cayley complexes

Quantum Tanner codes are a class of quantum low-density parity-check codes that provably display a linear minimum distance and a constant encoding rate in the asymptotic limit. When built from left--right Cayley complexes, they can be described through a lifting procedure and a base code, which we characterize. We also compute the dimension of quantum Tanner codes when the right degree of the complex is 2. Finally, we perform an extensive search over small groups and identify instances of quantum Tanner codes with parameters $[[144,12,11]]$, $[[432,20,\leq 22]]$ and $[[576,28,\leq 24]]$ for generators of weight 9.

quant-ph↗

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↗

arXiv2023

Analysis of the Error-Correcting Radius of a Renormalisation Decoder for Kitaev's 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 $d^{1/2}$ and 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.

quant-ph↗