Dequantizing quantum machine learning models using tensor networks

TL;DR

This paper uses tensor networks to analyze variational quantum machine learning models, identifying when they can be efficiently replaced by classical models, and introduces a classical kernel that matches the expressiveness of quantum kernels.

Contribution

It formalizes the dequantizability of VQML models using tensor networks and proposes a classical kernel that can replicate quantum kernel expressiveness.

Findings

VQML models can be characterized as a subclass of MPS models.

Conditions for dequantizability of VQML models are identified.

A classical kernel matching quantum kernel expressiveness is introduced.

Abstract

Ascertaining whether a classical model can efficiently replace a given quantum model -- dequantization -- is crucial in assessing the true potential of quantum algorithms. In this work, we introduced the dequantizability of the function class of variational quantum-machine-learning~(VQML) models by employing the tensor network formalism, effectively identifying every VQML model as a subclass of matrix product state (MPS) model characterized by constrained coefficient MPS and tensor product-based feature maps. From this formalism, we identify the conditions for which a VQML model's function class is dequantizable or not. Furthermore, we introduce an efficient quantum kernel-induced classical kernel which is as expressive as given any quantum kernel, hinting at a possible way to dequantize quantum kernel methods. This presents a thorough analysis of VQML models and demonstrates the versatility of our tensor-network formalism to properly distinguish VQML models according to their genuine quantum characteristics, thereby unifying classical and quantum machine-learning models within a single framework.