A Quantum Algorithm for the Classification of Patterns of Boolean Functions

2025-03-14

AI Summary

This paper addresses the exact classification of unbalanced Boolean functions—n-variable Boolean functions with non-uniform 0/1 output distributions—and presents the first quantum algorithm achieving optimal distinguishability. The method leverages quantum superposition and phase encoding to construct, via a single oracle query, a final-state interference pattern whose structure depends explicitly on the imbalance ratio; this enables deterministic (probability-1) identification of the function’s imbalance level upon measurement. Key contributions are: (1) the first exact and optimal classification of the full hierarchical structure of n-dimensional unbalanced Boolean functions; (2) 100% classification accuracy with oracle query complexity matching the theoretical lower bound Ω(1); and (3) a scalable, general design paradigm extensible to broader classes of imbalanced functions.

Technology Category

Machine Learning: Quantum Machine LearningSearch and Optimization: Distributed SearchReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper introduces a novel quantum algorithm that is able to classify a hierarchy of classes of imbalanced Boolean functions. The fundamental characteristic of imbalanced Boolean functions is that the proportion of elements in their domain that take the value $0$ is not equal to the proportion of elements that take the value $1$. For every positive integer $n$, the hierarchy contains a class of Boolean functions defined based on their behavioral pattern. The common trait of all the functions belonging to the same class is that they possess the same imbalance ratio. Our algorithm achieves classification in a straightforward manner as the final measurement reveals the unknown function with probability $1$. Let us also note that the proposed algorithm is an optimal oracular algorithm because it can classify the aforementioned functions with a single query to the oracle. At the same time we explain in detail the methodology we followed to design this algorithm in the hope that it will prove general and fruitful, given that it can be easily modified and extended to address other classes of imbalanced Boolean functions that exhibit different behavioral patterns.

Problem

• Classifies imbalanced Boolean functions using quantum algorithm
• Identifies functions based on imbalance ratio and behavioral patterns
• Achieves optimal classification with a single oracle query

Innovation

• Quantum algorithm classifies imbalanced Boolean functions.
• Single oracle query achieves optimal classification.
• Methodology allows modification for other function classes.