From Random Quantum Codes to Explicit qLDPC Codes via Local Properties
Abstract
Constructing explicit codes matching the parameters of random codes has been a central and largely elusive question in coding theory. The quantum setting is even more challenging since it is highly desirable that the quantum code be an LDPC code. Local coordinate-wise linear (LCL) [Levi, Mosheiff, and Shagrithaya, FOCS 2025] witnesses provide a unifying language for many coding-theoretic properties, from distance to list decoding and list recovery. In particular, it provides a framework to study properties of random linear codes, which achieve optimal parameters for many properties of linear codes. For CSS quantum codes, however, a local witness has two distinct ranks: its physical rank before quotienting by stabilizers and its logical rank after quotienting. We develop a quantum version of the LCL framework for nested spaces S ⊆ C, in which local constraints are imposed on physical representatives while independence is measured in the logical quotient C/S. The resulting theory gives a threshold theorem for random CSS codes, and as a consequence shows that the per-sector rate threshold is equal to the classical rate threshold. We also define a quantum analogue of subspace design [Guruswami and Xing, STOC 2013] and show that they can be described in a natural manner within the quantum-LCL framework. Finally, we give explicit constructions for arbitrary folded quantum-LCL properties, in a manner similar to the LCL derandomization of [Jeronimo and Shagrithaya, STOC 2026]. As a consequence, we obtain the first explicit constructions of quantum list-decodable codes and list-recoverable codes that have optimal list sizes, in addition to explicit quantum subspace design codes. We note that all our explicit constructions are qLDPC codes, an important property for quantum error-correcting codes.
Related
- Explicit Capacity-Achieving Quantum LDPC Codes List Decodable in Near-linear Time
- Thresholds for Local Coordinate-Wise Linear Properties of Random One-Point AG Codes
- High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices
- A Universal Threshold Theory for Local Properties of Random Linear Codes
- Lifted-Product QLDPC Codes in the Polynomial Domain
BibTeX
@misc{jeronimo2026from,
title = {{From Random Quantum Codes to Explicit qLDPC Codes via Local Properties}},
author = {Fernando Granha Jeronimo and Xiaojuan Ma and Nikhil Shagrithaya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40252},
eprint = {2609.40252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40252},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Random Quantum Codes to Explicit qLDPC Codes via Local Properties
AU - Fernando Granha Jeronimo
AU - Xiaojuan Ma
AU - Nikhil Shagrithaya
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40252
UR - https://arxiv.org/abs/2609.40252
AB - Constructing explicit codes matching the parameters of random codes has been a central and largely elusive question in coding theory. The quantum setting is even more challenging since it is highly desirable that the quantum code be an LDPC code. Local coordinate-wise linear (LCL) [Levi, Mosheiff, and Shagrithaya, FOCS 2025] witnesses provide a unifying language for many coding-theoretic properties, from distance to list decoding and list recovery. In particular, it provides a framework to study properties of random linear codes, which achieve optimal parameters for many properties of linear codes. For CSS quantum codes, however, a local witness has two distinct ranks: its physical rank before quotienting by stabilizers and its logical rank after quotienting. We develop a quantum version of the LCL framework for nested spaces S ⊆ C, in which local constraints are imposed on physical representatives while independence is measured in the logical quotient C/S. The resulting theory gives a threshold theorem for random CSS codes, and as a consequence shows that the per-sector rate threshold is equal to the classical rate threshold. We also define a quantum analogue of subspace design [Guruswami and Xing, STOC 2013] and show that they can be described in a natural manner within the quantum-LCL framework. Finally, we give explicit constructions for arbitrary folded quantum-LCL properties, in a manner similar to the LCL derandomization of [Jeronimo and Shagrithaya, STOC 2026]. As a consequence, we obtain the first explicit constructions of quantum list-decodable codes and list-recoverable codes that have optimal list sizes, in addition to explicit quantum subspace design codes. We note that all our explicit constructions are qLDPC codes, an important property for quantum error-correcting codes.
ER -