Learning quantum many-body systems from a few copies

Cambyse Rouzé1 and Daniel Stilck França2,3

1Department of Mathematics, Technische Universität München, 85748 Garching, Germany
2QMATH, Department of Mathematical Sciences, University of Copenhagen, Denmark
3Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France

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Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a state from a number of copies that scales polylogarithmically with the system's size and polynomially on the locality of the target observables. We show that this constitutes a provable exponential improvement in the number of copies over state-of-the-art tomography protocols. We achieve our results by combining the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The latter allows us to fine-tune the error made in estimating the expectation value of an observable in terms of how local it is and how well we approximate the expectation value of a fixed set of few-body observables. We conjecture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or outputs of shallow circuits. Moreover, we show improvements of the maximum entropy method beyond the sample complexity that are of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently as well as novel bounds on the condition number of covariance matrices of many-body states.

Traditionally, quantum tomography, the process of deriving a classical description of a quantum state from measurements, requires an exponential amount of data and computation as the size of the quantum system increases. This poses a significant challenge, especially as quantum systems grow more complex. However, this work shows that by focusing on more restricted classes of physically motivated states (Gibbs states) and only on physically relevant properties, it is possible to reduce the number of measurements doubly-exponentially when compared to traditional methods.

The new approach introduced in this work harnesses the power of maximum entropy methods combined with techniques from classical shadows and quantum optimal transport. Specifically, the new protocol can infer the values of all quasi-local observables of a state from a dramatically reduced number of copies that scale polylogarithmically with the system's size. This constitutes a major advancement over previous state-of-the-art tomography protocols, which often required a dauntingly large number of samples.

In practical terms, this means that scientists can now measure and predict the behavior of certain quantum systems with fewer resources.

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