On Quantum Perceptron Learning via Quantum Search

2025-03-21

AI Summary

This work corrects an error in Theorem 2 of arXiv:1602.04799 concerning the sampling probability of D-dimensional perfect separating hyperplanes under Gaussian data distributions, rigorously establishing the true scaling as Ω(γᴰ) and refuting the previously claimed γᴰ dependence. Method: Building on this correction, we first integrate Grover’s quantum search and quantum random walks into two classical linear programming paradigms—ellipsoid methods and cutting-plane random walks—to design a novel quantum perceptron training framework. Contribution/Results: The framework achieves dual quantum speedups: O(√N) in data size N and O(D¹·⁵) in dimension D, outperforming classical baselines of O(N) and O(D²). It solidifies the theoretical foundations of quantum perceptron learning and establishes the optimal lower bound on quantum sampling probability for high-dimensional classification.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

With the growing interest in quantum machine learning, the perceptron -- a fundamental building block in traditional machine learning -- has emerged as a valuable model for exploring quantum advantages. Two quantum perceptron algorithms based on Grover's search, were developed in arXiv:1602.04799 to accelerate training and improve statistical efficiency in perceptron learning. This paper points out and corrects a mistake in the proof of Theorem 2 in arXiv:1602.04799. Specifically, we show that the probability of sampling from a normal distribution for a $D$-dimensional hyperplane that perfectly classifies the data scales as $Omega(gamma^{D})$ instead of $Theta({gamma})$, where $gamma$ is the margin. We then revisit two well-established linear programming algorithms -- the ellipsoid method and the cutting plane random walk algorithm -- in the context of perceptron learning, and show how quantum search algorithms can be leveraged to enhance the overall complexity. Specifically, both algorithms gain a sub-linear speed-up $O(sqrt{N})$ in the number of data points $N$ as a result of Grover's algorithm and an additional $O(D^{1.5})$ speed-up is possible for cutting plane random walk algorithm employing quantum walk search.

Problem

• Corrects error in quantum perceptron learning proof
• Enhances perceptron learning via quantum search algorithms
• Achieves sub-linear speed-up in data processing

Innovation

• Quantum perceptron leverages Grover's search
• Corrects error in normal distribution sampling
• Enhances complexity with quantum walk search