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  • Open Access

Inequality constraints in variational quantum circuits with qudits

Alberto Bottarelli1,2,*, Sebastian Schmitt3, and Philipp Hauke1,2

  • *Contact author: alberto.bottarelli@untitn.it
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Phys. Rev. Research 7, 033202 – Published 29 August, 2025

DOI: https://doi.org/10.1103/3l96-41xf

Abstract

Quantum optimization is emerging as a prominent candidate for exploiting the capabilities of near-term quantum devices. Many application-relevant optimization tasks require the inclusion of inequality constraints, usually handled by enlarging the Hilbert space through the addition of slack variables. This approach, however, requires significant additional resources especially when considering multiple constraints. Here, we study an alternative direct implementation of these constraints within the quantum approximate optimization algorithm, achieved using qudit-sum gates, and compare it to the slack variable method generalized to qudits. We benchmark these approaches on three paradigmatic optimization problems. We find that the direct implementation of the inequality penalties vastly outperforms the slack variables method, especially when studying real-world inspired problems with many constraints. Within the direct penalty implementation, a linear energy penalty for unfeasible states outperforms other investigated functional forms, such as the canonical quadratic penalty. The proposed approach may thus be an enabling step for approaching realistic industry-scale and fundamental science problems with large numbers of inequality constraints.

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Phys. Rev. Research 7, 033202– Published 29 August, 2025
  • Received 18 October 2024
  • Accepted 25 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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