Provable learning separation for predicting time-evolution of quantum many-body systems

2026-07-07

AI Summary

This work investigates the efficient learning of unknown Hamiltonians governing quantum many-body systems from short-time evolution data, and rigorously characterizes the performance gap between quantum and classical machine learning in this setting. Framed within the PAC learning paradigm, the study formulates a supervised learning task that integrates Hamiltonian learning, Hamiltonian simulation, and classical shadow protocols for training and inference. Its central contribution is the first provable quantum–classical learning separation for a natural quantum machine learning problem rooted in physical dynamics: there exists a class of instances learnable by a quantum algorithm in polynomial time, yet provably intractable for any classical randomized algorithm unless BQP ⊆ P/poly. This result establishes a rigorous quantum advantage while preserving the physical interpretability and learnability of the underlying quantum system.

Abstract

Given that quantum computers are naturally suited to simulate the behavior of quantum many-body systems, an immediate question arises: can one formulate physically motivated quantum machine learning (QML) tasks that exhibit learning separations? We address this problem by studying the learnability of quantum many-body dynamics from the perspective of probably approximately correct (PAC)-learning. Concretely, we devise a supervised learning problem where the training set consists of specifications of randomized stabilizer probe states, evolution times sampled uniformly from a polynomially large time interval $[0,T]$, coupled with expectation values of certain observables evaluated on the resulting time-evolved state under an unknown Hamiltonian. For this learning task, we provide an efficient quantum procedure whose training phase learns the underlying Hamiltonian from short-time training samples, and whose deployment phase combines Hamiltonian simulation with the classical shadows protocol to perform inference on a newly given data point. By contrast, the existence of $O(\mathsf{poly}(n))$-time instances ensures classical hardness: by embedding a $\mathsf{BQP}$-complete computation into the polynomially long time-dynamics of a low-intersection variant of the Feynman-Kitaev clock Hamiltonian construction, we show that, for a certain family of input distributions, no randomized classical polynomial-time algorithm can fulfill our learning condition, unless $\mathsf{BQP}\subseteq\mathsf{P/poly}$. Furthermore, we show that the classically hard instance maintains quantum learnability. We also give an interpretation of our results in learning-assisted certified quantum simulation. Taken together, our results demonstrate a rigorous learning separation for a natural ML task based on Hamiltonian evolution, while building connections between quantum learning theory, quantum simulation, and QML.

Problem

• quantum many-body systems
• learning separation
• Hamiltonian evolution
• quant日晚间 simulation
• PAC-learning

Innovation

• learning separation
• quantum many-body dynamics
• PAC learning
• Hamiltonian simulation
• classical shadows