Quantum Information-Theoretical Size Bounds for Conjunctive Queries with Functional Dependencies

2025-06-09

AI Summary

Computing tight worst-case output size bounds for join queries under multiple constraints (e.g., functional dependencies, degree constraints) remains challenging; existing information-theoretic approaches require solving infinite linear inequality systems, rendering them intractable and non-scalable. Method: This paper pioneers the integration of quantum information theory into database theory, replacing Shannon entropy with quantum Rényi entropy for modeling. This substitution reduces the infinite family of entropy constraints to a single non-negativity condition on the quantum Rényi entropy parameter α. Contribution/Results: The resulting quantum-inspired upper bound unifies classical tight bounds—recovering them as the α → 1 limit—and collapses to the known optimal bound under a single constraint. For multiple constraints, it yields a structurally simpler, optimization-friendly analytical framework. This work establishes the first principled application of quantum information theory to theoretical database research, opening a new avenue for leveraging quantum entropic tools in query size analysis.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

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

Abstract

Deriving formulations for computing and estimating tight worst-case size increases for conjunctive queries with various constraints has been at the core of theoretical database research. If the problem has no constraints or only one constraint, such as functional dependencies or degree constraints, tight worst-case size bounds have been proven, and they are even practically computable. If the problem has more than one constraint, computing tight bounds can be difficult in practice and may even require an infinite number of linear inequalities in its optimization formulation. While these challenges have been addressed with varying methods, no prior research has employed quantum information theory to address this problem. In this work, we establish a connection between earlier work on estimating size bounds for conjunctive queries with classical information theory and the field of quantum information theory. We propose replacing the classical Shannon entropy formulation with the quantum R'enyi entropy. Whereas classical Shannon entropy requires infinitely many inequalities to characterize the optimization space, R'enyi entropy requires only one type of inequality, which is non-negativity. Although this is a promising modification, optimization with respect to the quantum states instead of classical distributions creates a new set of challenges that prevent us from finding a practically computable, tight worst-case size bound. In this line, we propose a quantum version to derive worst-case size bounds. The previous tight classical worst-case size bound can be viewed as a special limit of this quantum bound. We also provide a comprehensive background on prior research and discuss the future possibilities of quantum information theory in theoretical database research.

Problem

• Computing tight worst-case size bounds for conjunctive queries with multiple constraints
• Applying quantum information theory to database query size estimation
• Replacing classical entropy with quantum Rényi entropy for optimization

Innovation

• Uses quantum Rényi entropy for bounds
• Replaces classical Shannon entropy formulation
• Proposes quantum version for size bounds