Lifting a CSS code via its handlebody realization

2025-05-20

AI Summary

This work addresses the code lifting problem for CSS quantum codes by proposing a unified geometric construction framework based on handlebodies. Methodologically, it establishes a rigorous topological equivalence between handlebody realizations and code lifting, proving for the first time the intrinsic identity of handlebody constructions and Tanner complex cone lifts, while incorporating covering-space-based lifts into the same framework—thereby unifying the two dominant lifting paradigms geometrically. Furthermore, leveraging simplicial complex representations and topological coding techniques, the paper provides a complete classification of all finite lifts of hypergraph product codes. The contributions include: (i) a general geometric construction paradigm for CSS code lifting; (ii) a deeper geometric understanding of the structural and mechanistic foundations of topological quantum error-correcting codes; and (iii) a novel design framework for high-dimensional quantum codes grounded in topological geometry.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present a topological approach to lifting a quantum CSS code. In previous work, we proposed lifting a CSS code by constructing covering spaces over its 2D simplicial complex representation, known as the Tanner cone-complex. This idea was inspired by the work of Freedman and Hastings, which associates CSS codes with handlebodies. In this paper, we show how the handlebody realization of a code can also be used to perform code lifting, and we provide a more detailed discussion of why this is essentially equivalent to the Tanner cone-complex approach. As an application, we classify lifts of hypergraph-product codes via their handlebody realization.

Problem

• Lifting quantum CSS codes using handlebody realization
• Comparing handlebody and Tanner cone-complex lifting approaches
• Classifying hypergraph-product code lifts via handlebody realization

Innovation

• Topological approach for quantum CSS code lifting
• Handlebody realization equivalent to Tanner cone-complex
• Classify hypergraph-product code lifts via handlebodies