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Mitigating Barren Plateaus via Domain Decomposition in Variational Quantum Algorithms for Nonlinear PDEs

Laila S. Busaleh, Jeonghyeuk Kwon, Orlane Zang, Muhammad Hassan, Yvon Maday

Abstract

Barren plateaus present a major challenge in the training of variational quantum algorithms (VQAs), particularly for large-scale discretizations of nonlinear partial differential equations. In this work, we introduce a domain decomposition framework to mitigate barren plateaus by localizing the cost functional. Our strategy is based on partitioning the spatial domain into overlapping subdomains, each associated with a localized parameterized quantum circuit and measurement operator. Numerical results for the time-independent Gross-Pitaevskii equation show that the domain-decomposed formulation, allowing subdomain iterations to be interleaved with optimization iterations, exhibits improved solution accuracy and stable optimization compared to the global VQA formulation.

Mitigating Barren Plateaus via Domain Decomposition in Variational Quantum Algorithms for Nonlinear PDEs

Abstract

Barren plateaus present a major challenge in the training of variational quantum algorithms (VQAs), particularly for large-scale discretizations of nonlinear partial differential equations. In this work, we introduce a domain decomposition framework to mitigate barren plateaus by localizing the cost functional. Our strategy is based on partitioning the spatial domain into overlapping subdomains, each associated with a localized parameterized quantum circuit and measurement operator. Numerical results for the time-independent Gross-Pitaevskii equation show that the domain-decomposed formulation, allowing subdomain iterations to be interleaved with optimization iterations, exhibits improved solution accuracy and stable optimization compared to the global VQA formulation.

Paper Structure

This paper contains 21 sections, 2 theorems, 77 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Problem Formulation and Discretization Strategy
  2. Spectral Discretization
  3. Variational Quantum Formulation
  4. Full-Domain Behavior and Barren Plateaus
  5. Domain Decomposition Strategy
  6. Three-Subdomain Decomposition and Local Quantum Ansatz
  7. Sequential Subdomain Optimization
  8. Comparison with Classical Domain Decomposition
  9. Numerical Results
  10. Experimental Setup
  11. Controlled Subdomain Training to Avoid Prolonged Flat Regimes
  12. Comparison Between Full-Domain and Domain-Decomposed VQA
  13. Effect of Circuit Depth on the Optimization Dynamics
  14. Conclusion and Perspective
  15. Lie-algebraic perspective on barren plateaus and domain decomposition
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathfrak g_n$ be the dynamical Lie algebra generated by the full-domain ansatz defined in eq:appendix_dla. Assume that the entangling block $U_{\mathrm{ent}}$ consists of nearest-neighbour CNOT gates on a connected coupling graph. Then In particular,

Figures (9)

  • Figure 1: Hardware-efficient ansatz for $5$ qubits. Each entangling layer consists of single-qubit rotations followed by a ring of CNOT gates.
  • Figure 2: Full-domain training dynamics for $n=7,8,9$. Top left: energy error $|E-E_{\rm Newton}|$ relative to the Newton reference. Top-right: wavefunction error $\|\psi-\psi_N\|_2$. Bottom: Relative energy change between consecutive iterations for the full-domain formulation. The BFGS stopping tolerance is set to $10^{-20}$.
  • Figure 3: Three overlapping subdomains in index space. The rows show the indices belonging to the subdomains $\mathcal{I}_1$, $\mathcal{I}_2$, and $\mathcal{I}_3$. The bottom row displays the full grid index set, with colored markers highlighting the overlap regions.
  • Figure 4: Geometric illustration of three overlapping subdomains on the periodic domain $\Omega=[0,2\pi)$. Black points represent the discrete grid nodes, while the arcs indicate the three index subsets $\mathcal{I}_1$, $\mathcal{I}_2$, and $\mathcal{I}_3$.
  • Figure 5: Comparison of training dynamics between the classical domain decomposition and the VQA domain decomposition ($d=50$) for n=$7$.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Proposition 1
  • proof