Quantum Predictive Learning and Communication Complexity with Single Input

TL;DR

This paper introduces a new quantum learning model called Predictive Quantum (PQ), demonstrating an exponential separation from classical learning in relational concept classes and exploring quantum advantages in single-input communication complexity.

Contribution

The paper defines PQ, proves an exponential quantum-classical separation for relational classes, and analyzes quantum communication complexity in single-input mode, revealing new quantum advantages.

Findings

Efficient quantum learning of relational classes with exponential classical data requirements.

Unconditional quantum-classical separation in learning models.

Quantum communication complexity can exponentially surpass classical in single-input scenarios.

Abstract

We define a new model of quantum learning that we call Predictive Quantum (PQ). This is a quantum analogue of PAC, where during the testing phase the student is only required to answer a polynomial number of testing queries. We demonstrate a relational concept class that is efficiently learnable in PQ, while in any "reasonable" classical model exponential amount of training data would be required. This is the first unconditional separation between quantum and classical learning. We show that our separation is the best possible in several ways; in particular, there is no analogous result for a functional class, as well as for several weaker versions of quantum learning. In order to demonstrate tightness of our separation we consider a special case of one-way communication that we call single-input mode, where Bob receives no input. Somewhat surprisingly, this setting becomes nontrivial when relational communication tasks are considered. In particular, any problem with two-sided input can be transformed into a single-input relational problem of equal classical one-way cost. We show that the situation is different in the quantum case, where the same transformation can make the communication complexity exponentially larger. This happens if and only if the original problem has exponential gap between quantum and classical one-way communication costs. We believe that these auxiliary results might be of independent interest.