Solving the 3D Heat Equation with VQA via Remeshing-Based Warm Starts

Abstract: Quantum computing holds great promise for solving classically intractable problems such as linear systems and partial differential equations (PDEs). While fully fault-tolerant quantum computers remain out of reach, current noisy intermediate-scale quantum (NISQ) devices enable the exploration of hybrid quantum-classical algorithms. Among these, Variational Quantum Algorithms (VQAs) have emerged as a leading candidate for near-term applications. In this work, we investigate the use of VQAs to solve PDEs arising in stationary heat transfer. These problems are discretized via the finite element method (FEM), yielding linear systems of the form Ku=f, where K is the stiffness matrix. We define a cost function that encodes the thermal energy of the system, and optimize it using various ansatz families. To improve trainability and bypass barren plateaus, we introduce a remeshing strategy which gradually increases resolution by reusing optimized parameters from coarser discretizations. Our results demonstrate convergence of scalar quantities with mesh refinement. This work provides a practical methodology for applying VQAs to PDEs, offering insight into the capabilities and limitations of current quantum hardware.