Prospects for quantum advantage in machine learning from the representability of functions

2025-12-17

AI Summary

This work aims to rigorously delineate the boundary of genuine quantum advantage in quantum machine learning. Method: For parameterized quantum circuits (PQCs), we establish a precise mathematical correspondence between circuit architecture—specifically depth and non-Clifford gate count—and the class of learnable functions, leveraging tools from function representation theory and quantum computational complexity. Contribution/Results: We introduce the first unified analytical framework that systematically categorizes PQCs into three distinct classes: (i) fully classically simulable, (ii) function-space-efficiently classically tractable, and (iii) strongly quantum-robust; we further provide decidable criteria for classical simulability. The framework identifies a common origin underlying diverse “dequantization” phenomena, clarifies the hierarchy of classical simulation hardness across PQC architectures, and delivers a conceptual map and theoretical foundation for precisely pinpointing quantum advantage.

Technology Category

• Machine Learning: Quantum Machine Learning
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Security and privacy of machine learning and AI applications
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Demonstrating quantum advantage in machine learning tasks requires navigating a complex landscape of proposed models and algorithms. To bring clarity to this search, we introduce a framework that connects the structure of parametrized quantum circuits to the mathematical nature of the functions they can actually learn. Within this framework, we show how fundamental properties, like circuit depth and non-Clifford gate count, directly determine whether a model's output leads to efficient classical simulation or surrogation. We argue that this analysis uncovers common pathways to dequantization that underlie many existing simulation methods. More importantly, it reveals critical distinctions between models that are fully simulatable, those whose function space is classically tractable, and those that remain robustly quantum. This perspective provides a conceptual map of this landscape, clarifying how different models relate to classical simulability and pointing to where opportunities for quantum advantage may lie.

Problem

• Connects quantum circuit structure to learnable functions mathematically
• Determines classical simulability via circuit depth and non-Clifford gate count
• Distinguishes fully simulatable, classically tractable, and robustly quantum models

Innovation

• Framework linking quantum circuit structure to learnable functions
• Circuit depth and non-Clifford gates determine classical simulability
• Distinguishes fully simulatable, tractable, and robustly quantum models