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Semidefinite programming for quantum channel learning

Mikhail Gennadievich Belov*

Victor Victorovich Dubov† and Vadim Konstantinovich Ivanov‡

Alexander Yurievich Maslov§, Olga Vladimirovna Proshina∥, and Vladislav Gennadievich Malyshkin

  • Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, Moscow, Vorob'evy Gory 119991, Russia and Autretech Group, Skolkovo Innovation Center, Nobel Street, Building 7, Moscow 121205, Russia

  • *Contact author: mikhail.belov@tafs.pro
  • †Contact author: dubov@spbstu.ru
  • ‡Contact author: ivvadim@rambler.ru
  • §Contact author: maslov.ton@mail.ioffe.ru
  • ∥Contact author: proshina.ton@mail.ioffe.ru
  • Contact author: malyshki@ton.ioffe.ru
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Phys. Rev. E 114, 035302 – Published 10 September, 2026

DOI: https://doi.org/10.1103/kdl5-smrh

Abstract

The problem of reconstructing a quantum channel from a sample of classical data is considered. When the total fidelity can be represented as a ratio of two quadratic forms (e.g., in the case of mapping a mixed state to a pure state, projective operators, unitary learning, and others), semidefinite programming (sdp) can be applied to solve the fidelity optimization problem with respect to the Choi matrix. A remarkable feature of sdp is that the optimization is convex, which allows the problem to be efficiently solved by a variety of numerical algorithms. We have tested several commercially available sdp solvers, all of which allowed for the reconstruction of quantum channels of different forms. A notable feature is that the Kraus rank of the obtained quantum channel typically comprises less than a few percent of its maximal possible value. This suggests that a relatively small Kraus rank quantum channel is typically sufficient to describe experimentally observed classical data. The theory was also applied to the problem of reconstructing projective operators from data. Finally, we discuss a classical computational model based on quantum channel transformation, performed and calculated on a classical computer, possibly hardware-optimized.

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Phys. Rev. E 114, 035302– Published 10 September, 2026
  • Received 20 February 2026
  • Accepted 28 July 2026
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