Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2303.06755

Local Quantum Codes from Subdivided Manifolds

Elia Portnoy

Published Mar 12, 2023• 1 citation
arXivHTMLPDF

Abstract

For $n \ge 3$, we demonstrate the existence of quantum codes which are local in dimension $n$ with $V$ qubits, distance $V^{\frac{n-1}{n}}$, and dimension $V^{\frac{n-2}{n}}$, up to a $polylog(V)$ factor. The distance is optimal up to the polylog factor. The dimension is also optimal for this distance up to the polylog factor. The proof combines the existence of asymptotically good quantum codes, a procedure to build a manifold from a code by Freedman-Hastings, and a quantitative embedding theorem by Gromov-Guth.

DOI
10.48550/arxiv.2303.06755

Related papers

Geometrically Local Quantum and Classical Codes from Subdivision

Ting-Chun Lin, Adam Wills, Min-Hsiu Hsieh

A geometrically local quantum code is an error correcting code situated within $\mathbb{R}^D$, where the checks only act on qubits within a fixed spatial distance. The main question is: What is the optimal dimension and distance for a geometrically local code? Recently, Portnoy made a significant breakthrough with codes achieving optimal dimension and distance up to polylogs. However, the construction invokes a somewhat advanced mathematical result that involves lifting a chain complex to a manifold. This paper bypasses this step and streamlines the construction by noticing that a family of good quantum low-density parity-check codes, balanced product codes, naturally carries a two-dimensional structure. Together with a new embedding result that will be shown elsewhere, this quantum code achieves the optimal dimension and distance in all dimensions. In addition, we show that the code has an optimal energy barrier. We also discuss similar results for classical codes.

arXiv:2309.16104•Sep 28, 2023•1 citation

Towards local testability for quantum coding

Anthony Leverrier, Vivien Londe, Gilles Zémor

We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the p -faces of the n -cube (for n>p ) and stabilizer constraints with faces of dimension (p±1) . The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into N=2n−p−1(np) physical qubits and displays local testability with a soundness of Ω(1/log⁡(N)) beating the current state-of-the-art of 1/log2⁡(N) due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors.We then extend this code family by considering the quotient of the n -cube by arbitrary linear classical codes of length n . We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary n -cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.

arXiv:1911.03069•Quantum•Feb 24, 2022•17 citations

Almost linear decoder for optimal geometrically local quantum codes

Quinten Eggerickx, Adam Wills, Ting-Chun Lin +2 more

Geometrically local quantum codes, which are error-correction codes embedded in R D with checks acting only on qubits within a fixed spatial distance, have garnered significant interest. Recently, it has been demonstrated how to achieve geometrically local codes that maximize both the dimension and the distance, as well as the energy barrier of the code. In this work, we focus on the constructions involving subdivision, and we show that they have an almost linear time decoder, obtained by combining the decoder of the outer good qLDPC code and a generalized version of the Union-Find decoder. This provides the first decoder for an optimal geometrically local three-dimensional code. We demonstrate the existence of a finite threshold error rate under the code capacity noise model using a minimum weight perfect matching decoder. Furthermore, we argue that this threshold is also applicable to the decoder based on the generalized Union-Find algorithm.

arXiv:2411.02928•Physical Review Research•May 12, 2025•2 citations

Asymptotically Good Quantum Locally Testable Codes

William Gay, Fernando Granha Jeronimo

We construct explicit families of asymptotically good quantum locally testable codes over qubits. More precseily, we construct explicit quantum LDPC CSS codes over qubits having constant rate, constant relative distance and constant weight local testers with constant soundness.

quant-pharXiv:2609.20780•Sep 18, 2026•0 citations

The Quantum Hamming Bound in Arbitrary Local Dimension

Yu-Xuan Zhang, Jing-Ling Chen

The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension $q\ge3$. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For $q\ge4$, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.

quant-pharXiv:2606.22538•Jun 23, 2026•0 citations

Transform Arbitrary Good Quantum LDPC Codes into Good Geometrically Local Codes in Any Dimension

Xingjian Li, Ting-Chun Lin, Min-Hsiu Hsieh

Geometrically local quantum codes, comprised of qubits and checks embedded in $\mathbb{R}^D$ with local check operators, have been a subject of significant interest. A key challenge is identifying the optimal code construction that maximizes both dimension and distance. Recent advancements have produced several constructions, but these either depend on specific good quantum low-density parity-check (qLDPC) codes or are limited to three dimensions. In this work, we introduce a construction that can transform any good qLDPC code into an optimal geometrically local quantum code. Our approach hinges on a novel procedure that extracts a two-dimensional structure from an arbitrary three-term chain complex. We expect that this procedure will find broader applications in areas such as weight reduction and the geometric realization of chain complexes.

arXiv:2408.01769•Aug 3, 2024•0 citations
For n≥3n \ge 3n≥3, we demonstrate the existence of quantum codes which are local in dimension nnn with VVV qubits, distance Vn−1nV^{\frac{n-1}{n}}Vnn−1​, and dimension Vn−2nV^{\frac{n-2}{n}}Vnn−2​, up to a polylog(V)polylog(V)polylog(V) factor. The distance is optimal up to the polylog factor. The dimension is also optimal for this distance up to the polylog factor. The proof combines the existence of asymptotically good quantum codes, a procedure to build a manifold from a code by Freedman-Hastings, and a quantitative embedding theorem by Gromov-Guth.
A geometrically local quantum code is an error correcting code situated within RD\mathbb{R}^DRD, where the checks only act on qubits within a fixed spatial distance. The main question is: What is the optimal dimension and distance for a geometrically local code? Recently, Portnoy made a significant breakthrough with codes achieving optimal dimension and distance up to polylogs. However, the construction invokes a somewhat advanced mathematical result that involves lifting a chain complex to a manifold. This paper bypasses this step and streamlines the construction by noticing that a family of good quantum low-density parity-check codes, balanced product codes, naturally carries a two-dimensional structure. Together with a new embedding result that will be shown elsewhere, this quantum code achieves the optimal dimension and distance in all dimensions. In addition, we show that the code has an optimal energy barrier. We also discuss similar results for classical codes.
We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the p -faces of the n -cube (for n>p ) and stabilizer constraints with faces of dimension (p±1) . The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into N=2n−p−1(np) physical qubits and displays local testability with a soundness of Ω(1/log⁡(N)) beating the current state-of-the-art of 1/log2⁡(N) due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors.We then extend this code family by considering the quotient of the n -cube by arbitrary linear classical codes of length n . We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary n -cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.
Geometrically local quantum codes, which are error-correction codes embedded in R D with checks acting only on qubits within a fixed spatial distance, have garnered significant interest. Recently, it has been demonstrated how to achieve geometrically local codes that maximize both the dimension and the distance, as well as the energy barrier of the code. In this work, we focus on the constructions involving subdivision, and we show that they have an almost linear time decoder, obtained by combining the decoder of the outer good qLDPC code and a generalized version of the Union-Find decoder. This provides the first decoder for an optimal geometrically local three-dimensional code. We demonstrate the existence of a finite threshold error rate under the code capacity noise model using a minimum weight perfect matching decoder. Furthermore, we argue that this threshold is also applicable to the decoder based on the generalized Union-Find algorithm.
The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension q≥3q\ge3q≥3. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For q≥4q\ge4q≥4, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.
Geometrically local quantum codes, comprised of qubits and checks embedded in RD\mathbb{R}^DRD with local check operators, have been a subject of significant interest. A key challenge is identifying the optimal code construction that maximizes both dimension and distance. Recent advancements have produced several constructions, but these either depend on specific good quantum low-density parity-check (qLDPC) codes or are limited to three dimensions. In this work, we introduce a construction that can transform any good qLDPC code into an optimal geometrically local quantum code. Our approach hinges on a novel procedure that extracts a two-dimensional structure from an arbitrary three-term chain complex. We expect that this procedure will find broader applications in areas such as weight reduction and the geometric realization of chain complexes.