Jul 7, 2026 · quant-phJ/K move · Enter open · S save
Rahul Bandyopadhyay, Riccardo Molteni, Jens Eisert, Vedran Dunjko+1
⟨aQaL⟩, Applied Quantum Algorithms, Leiden University, The Netherlands · Leiden Institute of Advanced Computer Science (LIACS), Leiden University, Einsteinweg 55, 2333 CC, Leiden, The Netherlands · Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany · Fraunhofer Heinrich Hertz Institute, 10587 Berlin, Germany · Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany
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(poly(n))-time instances ensures classical hardness: by embedding a
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
BQP⊆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.