Sublinear quantum algorithms for training linear and kernel-based classifiers

TL;DR

This paper introduces sublinear quantum algorithms for training linear and kernel-based classifiers, achieving quadratic speedups over classical methods while maintaining classical outputs, with implications for near-term quantum machine learning.

Contribution

It presents the first sublinear quantum algorithms for classifier training with provable guarantees and minimal overhead, improving efficiency over classical algorithms.

Findings

Quantum algorithms run in rac{b1}{b1}b0(\u0000b7b7b7) time, quadratic speedup.

Algorithms produce classical outputs with minimal overhead.

Established tight lower bounds and discussed near-term implementation feasibility.

Abstract

We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training classifiers with constant margin runs in $\tilde{O}(n+d)$ time. We design sublinear quantum algorithms for the same task running in $\tilde{O}(\sqrt{n} +\sqrt{d})$ time, a quadratic improvement in both $n$ and $d$. Moreover, our algorithms use the standard quantization of the classical input and generate the same classical output, suggesting minimal overheads when used as subroutines for end-to-end applications. We also demonstrate a tight lower bound (up to poly-log factors) and discuss the possibility of implementation on near-term quantum machines. As a side result, we also give sublinear quantum algorithms for approximating the equilibria of $n$-dimensional matrix zero-sum games with optimal complexity $\tilde{\Theta}(\sqrt{n})$.