Reformulation of the No-Free-Lunch Theorem for Entangled Data Sets

TL;DR

This paper extends the no-free-lunch theorem to quantum machine learning, demonstrating that entanglement in training data influences learnability limits and can potentially violate classical bounds.

Contribution

It introduces a reformulated quantum NFL theorem that incorporates entanglement, showing entanglement as a key resource in quantum learning.

Findings

Entangled data sets can violate classical NFL limits.

The quantum NFL theorem accounts for entanglement, reducing learnability bounds.

Experimental validation on Rigetti's quantum computer supports the theoretical results.

Abstract

The no-free-lunch (NFL) theorem is a celebrated result in learning theory that limits one's ability to learn a function with a training data set. With the recent rise of quantum machine learning, it is natural to ask whether there is a quantum analog of the NFL theorem, which would restrict a quantum computer's ability to learn a unitary process (the quantum analog of a function) with quantum training data. However, in the quantum setting, the training data can possess entanglement, a strong correlation with no classical analog. In this work, we show that entangled data sets lead to an apparent violation of the (classical) NFL theorem. This motivates a reformulation that accounts for the degree of entanglement in the training set. As our main result, we prove a quantum NFL theorem whereby the fundamental limit on the learnability of a unitary is reduced by entanglement. We employ Rigetti's quantum computer to test both the classical and quantum NFL theorems. Our work establishes that entanglement is a commodity in quantum machine learning.