Machine Learning Approaches to Building Quantum Circuits for Sets of Matrices

Abstract

Machine Learning has become a useful instrument across many fields, but the opacity of typical ``black-box'' models limits their value for scientific discovery. In this paper we present a methodology that uses \emph{interpretable} Machine Learning to generate analytic mathematical hypotheses about a physical system, which can then be confirmed by a formal derivation. We demonstrate the methodology on the problem of decomposing unitary operators into elementary quantum gates. By training an interpretable model on circuits obtained numerically and analyzing its weights, we are led to a hypothesis for the analytic structure of the decomposition. Applied to the family of diagonal operators, this hypothesis yields a universal, analytic quantum circuit valid for an arbitrary number of qubits, whose CNOT count scales as 𝒪(2ⁿ), improving on the 𝒪(2.5ⁿ) of standard numerical methods. The resulting decomposition is proved rigorously in a companion paper.

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BibTeX

@misc{fedin2026machine,
  title = {{Machine Learning Approaches to Building Quantum Circuits for Sets of Matrices}},
  author = {Matvei Fedin and Andrey Morozov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.06633},
  eprint = {2605.06633},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.06633},
}

RIS

TY  - UNPB
TI  - Machine Learning Approaches to Building Quantum Circuits for Sets of Matrices
AU  - Matvei Fedin
AU  - Andrey Morozov
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2605.06633
UR  - https://arxiv.org/abs/2605.06633
AB  - Machine Learning has become a useful instrument across many fields, but the opacity of typical ``black-box'' models limits their value for scientific discovery. In this paper we present a methodology that uses \emph{interpretable} Machine Learning to generate analytic mathematical hypotheses about a physical system, which can then be confirmed by a formal derivation. We demonstrate the methodology on the problem of decomposing unitary operators into elementary quantum gates. By training an interpretable model on circuits obtained numerically and analyzing its weights, we are led to a hypothesis for the analytic structure of the decomposition. Applied to the family of diagonal operators, this hypothesis yields a universal, analytic quantum circuit valid for an arbitrary number of qubits, whose CNOT count scales as 𝒪(2^n), improving on the 𝒪(2.5^n) of standard numerical methods. The resulting decomposition is proved rigorously in a companion paper.
ER  -

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