On Quantum Perceptron Learning via Quantum Search

TL;DR

This work revisits quantum perceptron learning with Grover-based speedups and corrects a key probabilistic claim: the probability to sample a perfect classifier from a $D$-dimensional normal distribution scales as $Ω(γ^{D})$ rather than $Θ(γ)$. It then develops online quantum variants of the ellipsoid and cutting plane linear programming methods and introduces a cutting-plane quantum hit-and-run algorithm using Szegedy quantum walks, achieving sublinear speedups in the data size $N$ (up to $O(\sqrt{N})$) and, in some regimes, an additional $O(D^{1.5})$ factor. However, the exponential sampling cost in high dimensions substantially limits practical gains, highlighting a fundamental tension between quantum speedups and the curse of dimensionality in version-space sampling. The results provide a nuanced view of where quantum enhancements can help in perceptron learning and motivate exploring alternative quantum ML methods beyond the tested paradigms.

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 $Ω(γ^{D})$ instead of $Θ(γ)$, where $γ$ 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.

Figures (1)

• Figure 1: The margin and perfect sampling probability relationships (log scale). Given from mohri2018foundations, $\bm x_i'=(\underbrace{(-1)^i, \ldots,(-1)^i,(-1)^{i+1}}_{i \text{ first components }}, 0, \ldots, 0)$, $\bm x_i= \frac{\bm x_i'}{||\bm x_i'||}$ and $y_i=(-1)^{i+1}$, with its margin $\gamma$ (blue dotted line) upper bounded by $\sqrt{1/2^{D-1}}$ (cyan dotted line). Here sklearn.svm.LinearSVC is used to approximately calculate the margins.

Theorems & Definitions (6)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof