Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines

TL;DR

This paper demonstrates that variational quantum classifiers and quantum kernels can efficiently solve any BQP problem, indicating their potential for quantum advantage in machine learning tasks beyond classical capabilities.

Contribution

It proves the universal expressiveness of variational quantum classifiers and quantum kernels for solving all problems in the BQP complexity class.

Findings

Quantum classifiers can solve PromiseBQP-complete problems.

Existence of feature maps and kernels with quantum advantage.

Implication for quantum advantage in machine learning.

Abstract

Machine learning is considered to be one of the most promising applications of quantum computing. Therefore, the search for quantum advantage of the quantum analogues of machine learning models is a key research goal. Here, we show that variational quantum classifiers and support vector machines with quantum kernels can solve a classification problem based on the $k$-Forrelation problem, which is known to be PromiseBQP-complete. Because the PromiseBQP complexity class includes all Bounded-Error Quantum Polynomial-Time (BQP) decision problems, our results imply that there exists a feature map and a quantum kernel that make variational quantum classifiers and quantum kernel support vector machines efficient solvers for any BQP problem. Hence, this work implies that their feature map and quantum kernel, respectively, can be designed to have a quantum advantage for any classification problem that cannot be classically solved in polynomial time but contrariwise by a quantum computer.