Problem
Research questions and friction points this paper is trying to address.
explicit quantum codes
qLDPC codes
local coordinate-wise linear (LCL)
CSS quantum codes
quantum list decoding
From Random Quantum Codes to Explicit qLDPC Codes via Local Properties
🤖 AI Summary
This study addresses the fundamental challenge in coding theory of constructing explicit quantum LDPC codes that match the parameters of random codes. Building upon the locally checkable labeling (LCL) framework, this work develops a quantum LCL theory and derives a threshold theorem for random CSS codes. Explicit constructions are achieved through subspace design analogies, nested-space local constraints, logical quotient independence, and derandomization techniques. As a result, this research presents the first explicit quantum list-decodable codes, recoverable codes, and subspace design codes with optimal parameters, marking a critical breakthrough in the field.
📝 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 $S \subseteq C$, 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quantum LDPC codes
Local coordinate-wise linear (LCL)
CSS quantum codes
List decoding
Explicit constructions
🔎 Similar Papers
No similar papers found.
