Provable learning separation for predicting time-evolution of quantum many-body systems
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.
Authors 5
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Leiden University · University of Applied Sciences Leiden
Affiliation as printed
Leiden Institute of Advanced Computer Science (LIACS) , Leiden University , Einsteinweg 55 , 2333 CC , Leiden , The Netherlands
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Leiden University · University of Applied Sciences Leiden
Affiliation as printed
Leiden Institute of Advanced Computer Science (LIACS) , Leiden University , Einsteinweg 55 , 2333 CC , Leiden , The Netherlands
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Leiden University · University of Applied Sciences Leiden
Affiliation as printed
Leiden Institute of Advanced Computer Science (LIACS) , Leiden University , Einsteinweg 55 , 2333 CC , Leiden , The Netherlands
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Helmholtz-Zentrum Berlin für Materialien und Energie · Freie Universität Berlin
Affiliation as printed
Dahlem Center for Complex Quantum Systems , Freie Universität Berlin , 14195 Berlin , Germany
Helmholtz-Zentrum Berlin für Materialien und Energie , 14109 Berlin , Germany
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