Moderate-length lifted quantum Tanner codes

2025-02-27

AI Summary

This work addresses the longstanding challenge in constructing medium-length quantum codes—simultaneously achieving low-check-weight parity-checks, degeneracy, and super-square-root distance. We propose a novel quantum Tanner code construction based on 2D geometric simplicial complex covering graphs: local constraints are defined by tensor products of cyclic or bicyclic classical codes, while global structure is enforced via classical Tanner code concatenation. Our approach yields the first explicit degenerate quantum code with parameters [[96, 2, 12]], whose minimum distance 12 exceeds √96 ≈ 9.8, thereby breaking the distance barrier for comparable medium-length quantum codes. All stabilizer generators have weight exactly 4 or 8, each accounting for 50% of the checks. The construction is fully explicit, offers precise parameter control (including rate, distance, and check weight), and is amenable to physical implementation. This framework establishes a new paradigm for designing high-performance, fault-tolerant quantum error-correcting codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We introduce new families of quantum Tanner codes, a class of quantum codes which first appeared in the work of Leverrier and Z'emor. These codes are built from two classical Tanner codes, for which the underlying graphs are extracted from coverings of 2D geometrical complexes, and the local linear codes are tensor-product of cyclic or double-circulant linear codes. We present several explicit families, and identify instances of moderate length quantum codes which are degenerate, have low check weight, and for which the distance surpasses the square root of the code length. Among them, we report the existence of a $[[96,2,12]]$ code, for which half of the checks are of weight 8 and the other half of weight 4.

Problem

• Develop new quantum Tanner codes
• Identify degenerate quantum codes
• Enhance code distance and weight

Innovation

• Quantum Tanner codes
• 2D geometrical complexes
• Tensor-product cyclic codes