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

TL;DR

This paper proposes a domain decomposition approach to reduce barren plateaus in variational quantum algorithms for nonlinear PDEs, improving optimization stability and solution accuracy.

Contribution

It introduces a localized domain decomposition framework that partitions the problem into overlapping subdomains for more effective VQA training.

Findings

Enhanced solution accuracy with domain decomposition

Improved stability in optimization processes

Effective mitigation of barren plateaus in quantum algorithms

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.