On Quantum Perceptron Learning via Quantum Search

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 inarXiv:1602.04799to accelerate training and improve statistical efficiency in perceptron learning. This paper points out and corrects a mistake in the proof of Theorem 2 inarXiv: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.

@article{sun2025_2503.17308,
  title={ On Quantum Perceptron Learning via Quantum Search },
  author={ Xiaoyu Sun and Mathieu Roget and Giuseppe Di Molfetta and Hachem Kadri },
  journal={arXiv preprint arXiv:2503.17308},
  year={ 2025 }
}