A Quantum Algorithm for the Classification of Patterns of Boolean Functions

TL;DR

The paper addresses the problem of classifying a hierarchy of imbalanced Boolean functions using quantum computation, proposing the Boolean Function Pattern Quantum Classifier (BFPQC) as an exact, single‑query oracle classifier. It introduces pattern vectors, pattern bases $P_{2n}$, and an imbalance ratio $\rho_{2n}=\frac{1}{2}-\frac{1}{2^{n+1}}$, composing a framework that generalizes Deutsch–Jozsa-like problems to imbalanced patterns. A hierarchy of unitary classifiers $Q_{2n}$ with $Q_{2n}=Q_2^{\otimes n}$ enables the construction of QCPC$_{2n}$ circuits, which, given an oracle for a function $f\in F_{2n}$, output the index $i$ of $f$ in the computational basis with probability 1, i.e., a one‑shot classification. The approach is argued to be optimal in the oracle model and is designed to be adaptable to other imbalanced Boolean function classes, with potential implications for distributed quantum computing and quantum learning tasks.

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.

Figures (14)

• Figure 1: This figure shows the unitary transform $U_{ f }$, which is based on the oracle for the function $f$ and implements the standard schema \ref{['eq: Generic Unitary Transform U_f']}.
• Figure 2: This figure shows the unitary transform $U_{ f }$, again based on the oracle for the function $f$, but now implementing the schema \ref{['eq: Unitary Transform U_f']}.
• Figure 3: The duality between Boolean functions and their pattern vectors.
• Figure 4: This figure visualizes the initial segment of a quantum circuit that can be used for the classification of the functions in $F_{ 2 }$.
• Figure 5: This figure shows the quantum circuit that implements the unitary transform $Q_{ 2 }$ for the classification of the Boolean functions in $F_{ 2 }$.

Theorems & Definitions (11)

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