Cloning is as Hard as Learning for Stabilizer States

2026-04-16

AI Summary

This work investigates the relationship between the sample complexities of cloning and learning $n$-qubit stabilizer states, addressing the fundamental question of whether cloning is easier than learning. By establishing a connection between stabilizer state cloning and sample amplification of classical distributions with linear structure, and leveraging tools from representation theory, the abelian hidden subgroup framework, and a novel structured random purification channel, the authors prove for the first time that the optimal sample complexity for cloning is $\Theta(n)$. This result demonstrates that, even for highly structured quantum states, cloning and learning are equally demanding in terms of sample requirements. The finding not only deepens our understanding of the quantum no-cloning theorem but also reveals a profound link between quantum learning and the foundations of cryptography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The impossibility of simultaneously cloning non-orthogonal states lies at the foundations of quantum theory. Even when allowing for approximation errors, cloning an arbitrary unknown pure state requires as many initial copies as needed to fully learn the state. Rather than arbitrary unknown states, modern quantum learning theory often considers structured classes of states and exploits such structure to develop learning algorithms that outperform general-state tomography. This raises the question: How do the sample complexities of learning and cloning relate for such structured classes? We answer this question for an important class of states. Namely, for $n$-qubit stabilizer states, we show that the optimal sample complexity of cloning is $Θ(n)$. Thus, also for this structured class of states, cloning is as hard as learning. To prove these results, we use representation-theoretic tools in the recently proposed Abelian State Hidden Subgroup framework and a new structured version of the recently introduced random purification channel to relate stabilizer state cloning to a variant of the sample amplification problem for probability distributions that was recently introduced in classical learning theory. This allows us to obtain our cloning lower bounds by proving new sample amplification lower bounds for classes of distributions with an underlying linear structure. Our results provide a more fine-grained perspective on No-Cloning theorems, opening up connections from foundations to quantum learning theory and quantum cryptography.

Problem

• quantum cloning
• stabilizer states
• sample complexity
• quantum learning theory
• No-Cloning theorem

Innovation

• stabilizer states
• sample complexity
• quantum cloning
• representation theory
• sample amplification