Fragmentation is Efficiently Learnable by Quantum Neural Networks
This addresses a challenge in quantum machine learning for physicists by enabling efficient simulation of fragmented quantum systems, offering a rare example of a quantum learning task without classical dequantization.
The paper tackles the problem of efficiently learning the Schur transform for Hilbert space fragmentation using quantum neural networks, proving that gradient descent can achieve this with polynomial training data when fragmentation is strong, and demonstrates elimination of barren plateaus and poor local minima. The result shows no known efficient classical algorithms exist for this task, providing a quantum learning advantage.
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.