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

TL;DR

This paper introduces a novel quantum information-theoretic approach to derive tight worst-case size bounds for conjunctive queries with functional dependencies, potentially simplifying the characterization of these bounds.

Contribution

It proposes replacing classical Shannon entropy with quantum Rényi entropy to simplify the optimization process in size bounds computation for conjunctive queries.

Findings

Quantum Rényi entropy requires only non-negativity inequalities.

Classical bounds are a special case of the quantum bounds.

The approach opens new research directions in quantum information theory for databases.

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.