SDP bounds on quantum codes: rational certificates

2026-03-20

AI Summary

Determining the maximum size of quantum codes for given length and distance is a central problem in quantum coding theory, yet traditional numerical methods struggle to provide rigorous non-existence proofs. This work addresses this challenge by integrating semidefinite programming with rational infeasibility certificates to eliminate floating-point inaccuracies, thereby producing the first verifiable, rigorous upper bounds for multiple families of quantum codes with lengths ranging from 6 to 19. By combining a low-rank clustering solver with heuristic algebraic rounding techniques, the authors successfully convert numerical solutions into exact rational expressions. This approach improves upon the best-known upper bounds for 18 distinct n-qubit quantum codes and significantly enhances both the practicality and scalability of semidefinite programming in the study of quantum code bounds.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

A fundamental problem in quantum coding theory is to determine the maximum size of quantum codes of given block length and distance. A recent work introduced bounds based on semidefinite programming, strengthening the well-known quantum linear programming bounds. However, floating-point inaccuracies prevent the extraction of rigorous non-existence proofs from the numerical methods. Here, we address this by providing rational infeasibility certificates for a range of quantum codes. Using a clustered low-rank solver with heuristic rounding to algebraic expressions, we can improve upon $18$ upper bounds on the maximum size of $n$-qubit codes with $6 \leq n \leq 19$. Our work highlights the practicality and scalability of semidefinite programming for quantum coding bounds.

Problem

• quantum codes
• semidefinite programming
• code bounds
• infeasibility certificates
• quantum coding theory

Innovation

• semidefinite programming
• quantum codes
• rational certificates
• infeasibility certificates
• low-rank solver