Hybrid Quantum-Classical Approach for Solving Nonlinear PDEs Using Domain Decomposition Techniques to Mitigate Barren Plateaus

Y. Maday

Sponsor: ERC EMC2
Supervisor: Y. Maday

 

Background and Motivation

Parameterized quantum circuits (PQC) offer a promising alternative for the numerical resolution of complex partial differential equations (PDEs), particularly those exhibiting nonlinear behaviors such as the Burgers' equation or the nonlinear Gross–Pitaevskii equation. However, these approaches often suffer from the "barren plateau" effect, where gradients become exponentially small, thus limiting the effective training of deep quantum models.

The following articles propose innovative approaches to mitigate this effect:

  • [1] Proposes a unified theory explaining the origin of barren plateaus in deep quantum circuits.

  • [2] Introduces advanced techniques for localizing cost functionals by exploiting domain decomposition strategies.

  • [3] Describes hybrid quantum-classical methods to accelerate the convergence of quantum models applied to complex physical systems.

Objectives

  1. Adapt the methods proposed in these articles to design a hybrid quantum-classical algorithm for efficiently solving nonlinear PDEs.

  2. Implement domain decomposition strategies to localize cost functionals and reduce barren plateaus.

  3. Test this approach on classic PDEs such as the Burgers' equation or the nonlinear Gross–Pitaevskii equation, known for their chaotic and highly nonlinear characteristics.

References

[1] Marco Vinicio Sebastian Cerezo de la Roca, Michael Ragone, Bojko Bakalov, Martin Larocca, Frederic Antoine Sauvage, Alexander F Kemper, and Carlos Ortiz Marrero. A unified theory of barren plateaus for deep parametrized quantum circuits. Nature Communications, 15(LA-UR-23- 30483), 2024.
[2] https://arxiv.org/pdf/2407.17706v1
[3] https://arxiv.org/abs/2405.00781

Updated on April 16, 2025, 9:42 p.m.