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

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

TL;DR

The paper investigates the probabilistic link between quantum classification of Boolean functions and Hamming distance by introducing a pattern-basis framework and a Nearest Basis Ket Game. Through exhaustive enumeration for small pattern lengths and random sampling for larger lengths, it demonstrates that the classification probability generally decreases with Hamming distance, with structured, predictable exceptions in certain function classes. The authors delineate three distance subintervals based on the arity $n$, showing how the probability bound behaves within each region and highlighting spikes in specialized spaces such as $F_{Q_2}^{\otimes m}$. They provide a rigorous methodology and a practical diagnostic tool for assessing the reliability of quantum classification results, enabling both confident endorsements and principled dismissals of nearest-neighbor inferences in quantum algorithms.

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.

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

TL;DR

The paper investigates the probabilistic link between quantum classification of Boolean functions and Hamming distance by introducing a pattern-basis framework and a Nearest Basis Ket Game. Through exhaustive enumeration for small pattern lengths and random sampling for larger lengths, it demonstrates that the classification probability generally decreases with Hamming distance, with structured, predictable exceptions in certain function classes. The authors delineate three distance subintervals based on the arity , showing how the probability bound behaves within each region and highlighting spikes in specialized spaces such as . They provide a rigorous methodology and a practical diagnostic tool for assessing the reliability of quantum classification results, enabling both confident endorsements and principled dismissals of nearest-neighbor inferences in quantum algorithms.

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.
Paper Structure (11 sections, 4 theorems, 22 equations, 14 figures, 13 tables)

This paper contains 11 sections, 4 theorems, 22 equations, 14 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Preliminary concepts
  3. Perfectly classifiable Boolean functions
  4. What about nonclassifiable Boolean functions?
  5. What do experiments show?
  6. Experimental methodology
  7. The case of exhaustively enumerating all Boolean functions
  8. The case of sampling Boolean functions
  9. Interpretation of the experimental results
  10. Discussion and conclusions
  11. The Nearest Basis Ket Game

Key Result

proposition 1

Extended Function Product Realizes Pattern Vector Product Extended Function Product Realizes Pattern Vector Product If $f_{ \mathbf{ p } } \colon \mathbb{ B }^{ n } \rightarrow \mathbb{ B }$ and $g_{ \mathbf{ q } } \colon \mathbb{ B }^{ m } \rightarrow \mathbb{ B }$ are the Boolean functions realizi

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 bit vectors.
  • Figure 4: The above quantum circuit implements the pattern basis classification scheme.
  • Figure 5: This figure shows the quantum circuit that implements the $C_{ 2 }$ classifier.
  • ...and 9 more figures

Theorems & Definitions (21)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • definition 8
  • definition 9
  • definition 10
  • ...and 11 more