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Learning Mixed Quantum States in Large-Scale Experiments

Matteo Votto1,*, Marko Ljubotina2,3,4, Cécilia Lancien5, J. Ignacio Cirac6,3, Peter Zoller7,8, Maksym Serbyn4, Lorenzo Piroli9, and Benoît Vermersch1,10

  • *Contact author: votto@mit.edu
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Phys. Rev. Lett. 136, 090801 – Published 4 March, 2026

DOI: https://doi.org/10.1103/rbg2-f61m

Abstract

We present and test a protocol to learn the matrix-product operator (MPO) representation of an experimentally prepared quantum state. The protocol takes as input classical shadows corresponding to local randomized measurements, and outputs the tensors of an MPO maximizing a suitably defined fidelity with the experimental state. The tensor optimization is carried out sequentially, similarly to the well-known density matrix renormalization group algorithm. Our approach is provably efficient under certain technical conditions expected to be met in short-range correlated states and in typical noisy experimental settings. Under the same conditions, we also provide an efficient scheme to estimate fidelities between the learned and the experimental states. We experimentally demonstrate our protocol by learning entangled quantum states of up to N=96 qubits in a superconducting quantum processor. Our method upgrades classical shadows to large-scale quantum computation and simulation experiments.

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Phys. Rev. Lett. 136, 090801– Published 4 March, 2026
  • Received 25 July 2025
  • Revised 13 November 2025
  • Accepted 2 January 2026
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