Learning quantum many-body systems from a few copies

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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 scale 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.

OriginalsprogEngelsk
Artikelnummer1319
TidsskriftQuantum
Vol/bind8
Antal sider38
ISSN2521-327X
DOI
StatusUdgivet - 2024

Bibliografisk note

Funding Information:
DSF was supported by the VILLUM FONDEN via the QMATH Centre of Excellence under Grant No. 10059 and from the European Research Council (grant agreement no. 81876). The research of CR has been supported by project QTraj (ANR-20-CE40-0024-01) of the French National Research Agency (ANR) and by a Junior Researcher START Fellowship from the MCQST. DSF and CR are grateful to Richard Kueng, Fernando Brand\u00E3o and Giacomo De Palma for interesting discussions.

Publisher Copyright:
© 2024 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften. All rights reserved.

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