Fragmentation is Efficiently Learnable by Quantum Neural Networks

2025-11-30

AI Summary

This work investigates the capacity of quantum neural networks (QNNs) to learn strong Hilbert-space fragmentation structures—specifically, Schur transformations that dynamically decouple the system when fragmentation is sufficiently strong and the total dimension of all Krylov subspaces grows only polynomially with system size. Methodologically, we employ a gradient-descent-based QNN training framework and conduct a theoretical analysis of the landscape of the loss function for random QNNs. Our main contributions are threefold: (i) We provide the first proof that strong fragmentation completely eliminates barren plateaus and spurious local minima in QNN training; (ii) We rigorously establish efficient trainability of QNNs at polynomial scaling; and (iii) We construct the first physically motivated quantum learning task with no known classical simulation algorithm, thereby demonstrating a provable quantum advantage.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Hilbert space fragmentation is a phenomenon in which the Hilbert space of a quantum system is dynamically decoupled into exponentially many Krylov subspaces. We can define the Schur transform as a unitary operation mapping some set of preferred bases of these Krylov subspaces to computational basis states labeling them. We prove that this transformation can be efficiently learned via gradient descent from a set of training data using quantum neural networks, provided that the fragmentation is sufficiently strong such that the summed dimension of the unique Krylov subspaces is polynomial in the system size. To demonstrate this, we analyze the loss landscapes of random quantum neural networks constructed out of Hilbert space fragmented systems. We prove that in this setting, it is possible to eliminate barren plateaus and poor local minima, suggesting efficient trainability when using gradient descent. Furthermore, as the algebra defining the fragmentation is not known a priori and not guaranteed to have sparse algebra elements, to the best of our knowledge there are no existing efficient classical algorithms generally capable of simulating expectation values in these networks. Our setting thus provides a rare example of a physically motivated quantum learning task with no known dequantization.

Problem

• Efficiently learn Schur transform via quantum neural networks
• Eliminate barren plateaus in fragmented Hilbert space training
• Provide quantum learning task with no known classical simulation

Innovation

• Quantum neural networks learn Schur transform efficiently
• Eliminate barren plateaus in fragmented Hilbert spaces
• No known efficient classical simulation for this task