[Paper Review] The Quantum Version Of Classification Decision Tree Constructing Algorithm C5.0

Kamil Khadiev, Ilnaz Mannapov+1 more|arXiv (Cornell University)|Jul 16, 2019

Quantum Computing Algorithms and ArchitectureComputer Science22 references18 citations

TL;DR

This paper proposes a quantum-enhanced version of the C5.0 decision tree algorithm, leveraging quantum subroutines like amplitude amplification and Dürr-Høyer minimum search to achieve a speedup. It improves the classical C5.0 algorithm using self-balancing binary search trees and presents a quantum version (QC5.0) with a running time of $ O(h\sqrt{d}\log d \cdot N\log N) $, offering nearly quadratic speedup over classical methods in the number of attributes.

ABSTRACT

In the paper, we focus on complexity of C5.0 algorithm for constructing decision tree classifier that is the models for the classification problem from machine learning. In classical case the decision tree is constructed in $O(hd(NM+N \log N))$ running time, where $M$ is a number of classes, $N$ is the size of a training data set, $d$ is a number of attributes of each element, $h$ is a tree height. Firstly, we improved the classical version, the running time of the new version is $O(h\cdot d\cdot N\log N)$. Secondly, we suggest a quantum version of this algorithm, which uses quantum subroutines like the amplitude amplification and the D{ü}rr-Høyer minimum search algorithms that are based on Grover's algorithm. The running time of the quantum algorithm is $O\big(h\cdot \sqrt{d}\log d \cdot N \log N\big)$ that is better than complexity of the classical algorithm.

Motivation & Objective

• To improve the classical C5.0 algorithm's efficiency by reducing its time complexity using self-balancing binary search trees.
• To develop a quantum version of the C5.0 algorithm that leverages quantum subroutines for faster computation.
• To achieve a significant speedup in constructing decision trees for large-scale classification problems.
• To demonstrate the feasibility and performance gain of quantum machine learning in decision tree construction.

Proposed method

• Replacing classical data structures with self-balancing binary search trees (e.g., AVL or Red-Black trees) to optimize attribute value tracking and reduce sorting overhead.
• Using the Dürr-Høyer quantum algorithm for minimum search to accelerate the selection of optimal split attributes in the decision tree.
• Applying amplitude amplification to enhance the probability of finding the best gain ratio among attributes in superposition.
• Integrating quantum subroutines into a hybrid classical-quantum framework where quantum routines handle attribute selection, and classical logic manages tree construction.
• Repeating the quantum maximum-finding procedure $ \log d $ times to reduce error probability and improve reliability of split selection.
• Using a query model where quantum algorithms access training data via black-box oracles, measuring runtime by number of queries.

Experimental results

Research questions

• RQ1 Can the classical C5.0 algorithm be optimized using self-balancing binary search trees to reduce time complexity?
• RQ2 Can quantum algorithms be effectively applied to the attribute selection phase of decision tree construction?
• RQ3 What is the theoretical running time improvement of a quantum-enhanced C5.0 algorithm compared to its classical counterpart?
• RQ4 How does the success probability of quantum attribute selection scale with the number of internal nodes in the tree?
• RQ5 Can the quantum version achieve a near-quadratic speedup in the number of attributes?

Key findings

• The improved classical C5.0 algorithm using self-balancing binary search trees achieves a running time of $ O(h d N \log N) $, reducing the original complexity by eliminating redundant sorting steps.
• The quantum version, QC5.0, achieves a running time of $ O(h \sqrt{d} \log d \cdot N \log N) $, offering a nearly quadratic speedup in the number of attributes $ d $.
• The success probability of QC5.0 is $ O\big{(}(1 - \frac{1}{d})^k\big{)} $, where $ k $ is the number of internal nodes, indicating high reliability when repeated $ \log d $ times.
• The quantum speedup is achieved by combining amplitude amplification and Dürr-Høyer’s minimum-finding algorithm to efficiently identify the best split attribute.
• The quantum subroutine reduces the time complexity of the attribute selection step from $ O(d) $ to $ O(\sqrt{d}) $, leading to overall performance gains on large datasets.
• The theoretical analysis confirms that QC5.0 provides a significant asymptotic improvement over classical C5.0, especially in high-dimensional data settings.