A Systematic Approach to Hyperbolic Quantum Error Correction Codes

2025-04-10

AI Summary

Conventional Euclidean-space quantum error-correcting codes (QECCs) suffer from low encoding rates and limited fault-tolerance thresholds. Method: This work systematically constructs hyperbolic-space QECCs (HQECCs) for the first time, embedding qubits into hyperbolic tessellations to exploit their exponential graph expansion. We propose a general constructive framework for HQECCs, together with novel algorithms for face-cycle enumeration and logical operator computation, enabling scalable parametric analysis. Integrating hyperbolic crystallography, topological coding theory, and graph-theoretic algorithms, we model and numerically simulate HQECCs based on two families of hyperbolic tessellations. Results: HQECCs achieve significantly higher encoding rates, improved error thresholds, and tunable code distances—establishing a new paradigm and verifiable foundation for high-dimensional non-Euclidean quantum fault-tolerant architectures.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Hyperbolic quantum error correction codes (HQECCs) leverage the unique geometric properties of hyperbolic space to enhance the capabilities and performance of quantum error correction. By embedding qubits in hyperbolic lattices, HQECCs achieve higher encoding rates and improved error thresholds compared to conventional Euclidean codes. Building on recent advances in hyperbolic crystallography, we present a systematic framework for constructing HQECCs. As a key component of this framework, we develop a novel algorithm for computing all plaquette cycles and logical operators associated with a given HQECC. To demonstrate the effectiveness of this approach, we utilize this framework to simulate two HQECCs based respectively on two relevant examples of hyperbolic tilings. In the process, we evaluate key code parameters such as encoding rate, error threshold, and code distance for different sub-lattices. This work establishes a solid foundation for a systematic and comprehensive analysis of HQECCs, paving the way for the practical implementation of HQECCs in the pursuit of robust quantum error correction strategies.

Problem

• Enhancing quantum error correction using hyperbolic space geometry
• Developing systematic framework for hyperbolic quantum error correction codes
• Evaluating key parameters like encoding rate and error threshold

Innovation

• Embedding qubits in hyperbolic lattices
• Novel algorithm for plaquette cycles computation
• Systematic framework for HQECC construction