Learning the Hamiltonian that describes interactions in a quantum system is an important task in both condensed-matter physics and the verification of quantum technologies. Its classical analogue arises as a central problem in machine learning known as learning Boltzmann machines. Previously, the best known methods for quantum Hamiltonian learning with provable performance guarantees required a number of measurements that scaled exponentially with the number of particles. Here we prove that only a polynomial number of local measurements on the thermal state of a quantum system are necessary and sufficient for accurately learning its Hamiltonian. We achieve this by establishing that the absolute value of the finite-temperature free energy of quantum many-body systems is strongly convex with respect to the interaction coefficients. The framework introduced in our work provides a theoretical foundation for applying machine learning techniques to quantum Hamiltonian learning, achieving a long-sought goal in quantum statistical learning.
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The data presented in the figures are available at https://github.com/gitmehdis/hamiltonian-learning.
The codes used to generate the figures are available at https://github.com/gitmehdis/hamiltonian-learning.
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We thank A. Harrow, Y. Huang, R. La Placa, S. Subramanian, J. Wright and H. Yuen for helpful discussions. Part of this work was done when S.A. and T.K. were visiting Perimeter Institute. S.A. was supported in part by the Army Research Laboratory and the Army Research Office under grant no. W911NF-20-1-0014. The work was done when A.A. was affiliated to the Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo and the Perimeter Institute for Theoretical Physics. A.A. was supported by the Canadian Institute for Advanced Research, through funding provided to the Institute for Quantum Computing by the Government of Canada and the Province of Ontario. Perimeter Institute is also supported in part by the Government of Canada and the Province of Ontario. T.K. was supported by the RIKEN Center for AIP and JSPS KAKENHI grant no. 18K13475. M.S. was supported by NSF grant no. CCF-1729369, a Samsung Advanced Institute of Technology Global Research Cluster and grant no. FXQi-RFP-1811A from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation.
The authors declare no competing interests.
Peer review information Nature Physics thanks Vedran Dunjko and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Anshu, A., Arunachalam, S., Kuwahara, T. et al. Sample-efficient learning of interacting quantum systems. Nat. Phys. 17, 931–935 (2021). https://doi.org/10.1038/s41567-021-01232-0