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arXiv:2510.12736 · Stats

Probabilistic Links Between Quantum Classification of Patterns of Boolean Functions and Hamming Distance

Theodore Andronikos, Constantinos Bitsakos, Konstantinos Nikas, Georgios Goumas, Nectarios Koziris

Published Jan 1, 2026• 2 citations
arXivHTMLPDF

Abstract

This article investigates the probabilistic relationship between quantum classification of Boolean functions and their Hamming distance. By integrating concepts from quantum computing, information theory, and combinatorics, we explore how Hamming distance serves as a metric for analyzing deviations in function classification. Our extensive experimental results confirm that the Hamming distance is a pivotal metric for validating nearest neighbors in the process of classifying random functions. One of the significant conclusions we arrived is that the successful classification probability decreases monotonically with the Hamming distance. However, key exceptions were found in specific classes, revealing intra-class heterogeneity. We have established that these deviations are not random but are systemic and predictable. Furthermore, we were able to quantify these irregularities, turning potential errors into manageable phenomena. The most important novelty of this work is the demarcation, for the first time to the best of our knowledge, of precise Hamming distance intervals for the classification probability. These intervals bound the possible values the probability can assume, and provide a new foundational tool for probabilistic assessment in quantum classification. Practitioners can now endorse classification results with high certainty or dismiss them with confidence. This framework can significantly enhance any quantum classification algorithm’s reliability and decision-making capability.

DOI
10.3390/stats9010005

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This article investigates the probabilistic relationship between quantum classification of Boolean functions and their Hamming distance. By integrating concepts from quantum computing, information theory, and combinatorics, we explore how Hamming distance serves as a metric for analyzing deviations in function classification. Our extensive experimental results confirm that the Hamming distance is a pivotal metric for validating nearest neighbors in the process of classifying random functions. One of the significant conclusions we arrived is that the successful classification probability decreases monotonically with the Hamming distance. However, key exceptions were found in specific classes, revealing intra-class heterogeneity. We have established that these deviations are not random but are systemic and predictable. Furthermore, we were able to quantify these irregularities, turning potential errors into manageable phenomena. The most important novelty of this work is the demarcation, for the first time to the best of our knowledge, of precise Hamming distance intervals for the classification probability. These intervals bound the possible values the probability can assume, and provide a new foundational tool for probabilistic assessment in quantum classification. Practitioners can now endorse classification results with high certainty or dismiss them with confidence. This framework can significantly enhance any quantum classification algorithm’s reliability and decision-making capability.
This paper studies the important problem of quantum classification of Boolean functions from an entirely novel perspective. Typically, quantum classification algorithms allow us to classify functions with a probability of 1.0, if we are promised that they meet specific unique properties. The primary objective of this study is to explore whether it is feasible to obtain any insights when the input function deviates from the promised class. For concreteness, we use a recently introduced quantum algorithm that is designed to classify a large class of imbalanced Boolean functions with probability 1.0 using just a single oracular query. First, we establish a completely new concept characterizing “nearness” between Boolean functions. Utilizing this concept, we show that, as long as the unknown function is close enough to the promised class, it is still possible to obtain useful information about its behavioral pattern from the classification algorithm. In this regard, the current study is among the first to provide evidence that shows how useful it is to apply quantum classification algorithms to functions outside the promised class in order to get a glimpse of important information.
We present a novel quantum algorithm to evaluate the hamming distance between two unknown oracles via measuring the degree of entanglement between two ancillary qubits. In particular, we use the power of the entanglement degree based quantum computing model that preserves at most the locality of interactions within the quantum model structure. This model uses one of two techniques to retrieve the solution of a quantum computing problem at hand. In the first technique, the solution of the problem is obtained based on whether there is an entanglement between the two ancillary qubits or not. In the second, the solution of the quantum computing problem is obtained as a function in the concurrence value, and the number of states that can be generated from the Boolean variables. The proposed algorithm receives two oracles, each oracle represents an unknown Boolean function, then it measures the hamming distance between these two oracles. The hamming distance is evaluated based on the second technique. It is shown that the proposed algorithm provides exponential speedup compared with the classical counterpart for Boolean functions that have large numbers of Boolean variables. The proposed algorithm is explained via a case study. Finally, employing recently developed experimental techniques, the proposed algorithm has been verified using IBM's quantum computer simulator.
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 n-ary Boolean functions defined according to 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 a probability of 1.0. Let us also note that the proposed algorithm is an optimal oracular algorithm because it can classify the aforementioned functions with just 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.
The advancement of technology in Quantum Computing has brought possibilities for the execution of algorithms in real quantum devices. However, the existing errors in the current quantum hardware and the low number of available qubits make it necessary to use solutions that use fewer qubits and fewer operations, mitigating such obstacles. Hadamard Classifier (HC) is a distance-based quantum machine learning model for pattern recognition. We present a new classifier based on HC named Quantum One-class Classifier (QOCC) that consists of a minimal quantum machine learning model with fewer operations and qubits, thus being able to mitigate errors from NISQ (Noisy Intermediate-Scale Quantum) computers. Experimental results were obtained by running the proposed classifier on a quantum device and show that QOCC has advantages over HC.
Quantum logic reversible synthesis is a fundamental operation in quantum computing. One of the most challenging issues in this field resides in navigating the immense search space to synthesize the most compact circuit configurations, which are critical for realizing reliable, noise-free, and error-free quantum computing systems. To address this challenge, this study proposes a novel hypercube-encoded quantum-inspired optimization framework to formulate the synthesis task as a trajectory-finding process. This structure-informed domain knowledge transformation delivers exceptional search direction guidance, moving away from blind, black-box exploration. Specifically, by mapping the reversible functions onto the hypercube architecture, the framework embeds explicit dual Hamming-distance (HMD) guidance metrics into a global-best guided quantum-inspired tabu search (GQTS) engine. To minimize computational cost and enhance search efficiency, the framework incorporates a domain-informed initialization and couples a streamlined two-particle configuration with a global-best mechanism, thereby amplifying the efficiency of the underlying quantum-inspired updating mechanism to escape local optima and rapidly converge once a successfully synthesized superior path is locked. Under an online step relaxation mechanism, the framework preserves exceptional structural optimization flexibility without altering the underlying hypercube representation. The framework’s significance is rigorously evaluated against both rigid hypercube rule-based methods and generic randomized search-based heuristics, using gate count and exact-optimality as primary evaluation metrics. Extensive ablation studies first validate the individual and synergistic contributions of each core algorithmic component. Experimental results demonstrate that for the complete set of 3-bit reversible functions, the proposed HMD-guided GQTS (HMD-GQTS) framework achieves over 98% exact-optimal circuits in a single fine-tuned sweep, with selective retries attaining 100% exhaustive optimal coverage. Furthermore, for typical 4-bit benchmark instances, the method consistently delivers competitive gate counts, matching or improving upon previous hypercube-based outcomes. Through the seamless integration of structure-informed guidance and coordinated optimization mechanisms, the framework preserves exceptional structural flexibility and algorithmic robustness, offering an effective, low-cost, and highly scalable avenue for reversible circuit synthesis.