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Universal resources for quantum approximate optimization algorithm and quantum annealing

Pablo Díez-Valle1,2,*, Fernando J. Gómez-Ruiz3,†, Diego Porras1,‡, and Juan José García-Ripoll1,§

  • 1Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain
  • 2Instituto Tecnológico de Galicia, Cantón Grande 9, Planta 3, 15003 A Coruña, Spain
  • 3Departamento de Física, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain

  • *Contact author: pvalle@itg.es
  • †Contact author: fegomezr@fis.uc3m.es
  • ‡Contact author: diego.porras@csic.es
  • §Contact author: juanjose.garcia.ripoll@csic.es
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Phys. Rev. Research 8, 013211 – Published 24 February, 2026

DOI: https://doi.org/10.1103/hxv2-sbr7

Abstract

The quantum approximate optimization algorithm (QAOA) is a variational ansatz that resembles the Trotterized dynamics of a quantum annealing (QA) protocol. This work formalizes this connection formally and empirically, showing that the angles of a multilayer QAOA circuit converge to universal QA trajectories. Furthermore, the errors in both QAOA circuits and QA paths act as thermal excitations in pseudo-Boltzmann probability distributions whose temperature decreases with the invested resource—i.e., integrated angles or total time—and which in QAOA also contain a higher temperature arising from the Trotterization. This also means that QAOA and QA are cooling protocols and simulators of partition functions whose target temperature can be tuned by rescaling the universal trajectory. The average cooling power of both methods exhibits favorable algebraic scalings with respect to the target temperature and problem size, whereby in QAOA the coldest temperature is inversely proportional to the number of layers, T∼1/p, and to the integrated angles—or integrated interactions in QA.

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Phys. Rev. Research 8, 013211– Published 24 February, 2026
  • Received 20 June 2025
  • Revised 16 July 2025
  • Accepted 22 January 2026
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