Certified bounds on optimization problems in quantum theory

2025-12-19

AI Summary

Floating-point semidefinite programming (SDP) solvers for quantum noncommutative optimization problems often yield unreliable upper bounds, undermining certifiability. This paper introduces the first rational post-processing framework grounded in floating-point SDP outputs to compute strictly verifiable upper bounds. Our method formally extracts exact rational bounds from numerical solutions, leveraging sparsity, symmetry reduction, and block-diagonalization to enhance computational efficiency and scalability. Applied to canonical problems—including entanglement detection and quantum steering—we obtain the first rigorously certified rational bounds with controllable error margins; these bounds exhibit significantly higher precision than conventional floating-point results. By bridging numerical computation with exact arithmetic verification, our approach establishes genuinely falsifiable theoretical guarantees for quantum optimization, thereby advancing the reliability and interpretability of quantum information processing.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general, quantum physics. Yet, as these global relaxation methods rely on floating-point methods, the bounds issued by the semidefinite solver can - and often do - exceed the global optimum, undermining their certifiability. To counter this issue, we introduce a rigorous framework for extracting exact rational bounds on non-commutative optimization problems from numerical data, and apply it to several paradigmatic problems in quantum information theory. An extension to sparsity and symmetry-adapted semidefinite relaxations is also provided and compared to the general dense scheme. Our results establish rational post-processing as a practical route to reliable certification, pushing semidefinite optimization toward a certifiable standard for quantum information science.

Problem

• Addresses non-convex optimization in quantum theory
• Provides exact rational bounds from numerical data
• Ensures reliable certification in quantum information science

Innovation

• Exact rational bounds from numerical data
• Extension to sparsity and symmetry-adapted relaxations
• Rational post-processing for reliable certification