Breaking the Orthogonality Barrier in Quantum LDPC Codes

2026-01-13

AI Summary

This work addresses the longstanding challenge in quantum low-density parity-check (LDPC) code design—namely, the tension between orthogonality constraints and regular structure, which typically prevents simultaneous achievement of high girth and large minimum distance. The authors propose a novel construction that employs permutation matrices with controlled commutativity and enforces local orthogonality only at critical positions, thereby preserving matrix regularity while circumventing conventional performance trade-offs. The resulting (3,12)-regular quantum LDPC code, denoted [[9216,4612,≤48]], achieves a girth of 8 and is free from structural limitations on its minimum distance upper bound. Under a physical error rate of 4%, the combination of belief propagation decoding with low-complexity post-processing reduces the frame error rate to 10⁻⁸.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular $[[9216,4612, \leq 48]]$ quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as $10^{-8}$ on the depolarizing channel with error probability $4 \%$.

Problem

• quantum LDPC codes
• orthogonality constraints
• girth
• minimum distance
• regularity

Innovation

• quantum LDPC codes
• orthogonality constraints
• girth
• permutation matrices
• belief-propagation decoding