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

Samuel Donachie, Ulysse Remond, Arthur Mathorel, Kyryl Kazymyrenko

TL;DR

The paper tackles solving the 3D heat equation $\Delta u = f$ discretized by FEM into $K u = f$ using variational quantum algorithms on NISQ devices. It introduces a physics‑informed cost and a remeshing cascade warm‑start strategy that reuses optimized solutions from coarser meshes to initialize larger problems, improving trainability and convergence. In noiseless simulations, it compares multiple ansätze and initialization schemes, showing that Cascade warm starts enable scalable convergence up to 15 qubits, whereas cold starts stall or diverge. The work also proposes a hardware benchmarking framework by comparing ideal, noisy, and stochastic results against the PDE target, highlighting both potential gains and current hardware limitations for PDEsolving with quantum computers.

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.

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

TL;DR

The paper tackles solving the 3D heat equation discretized by FEM into using variational quantum algorithms on NISQ devices. It introduces a physics‑informed cost and a remeshing cascade warm‑start strategy that reuses optimized solutions from coarser meshes to initialize larger problems, improving trainability and convergence. In noiseless simulations, it compares multiple ansätze and initialization schemes, showing that Cascade warm starts enable scalable convergence up to 15 qubits, whereas cold starts stall or diverge. The work also proposes a hardware benchmarking framework by comparing ideal, noisy, and stochastic results against the PDE target, highlighting both potential gains and current hardware limitations for PDEsolving with quantum computers.

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.
Paper Structure (15 sections, 9 equations, 5 figures, 8 tables)

This paper contains 15 sections, 9 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Problem and Method
  3. Problem definition
  4. The choice of the Ansatz
  5. Results
  6. Cold start
  7. Uniform start
  8. The Cascade Protocol
  9. Conclusion
  10. Acknowledgments.
  11. Disclosure of Interests.
  12. Using our results as a Benchmark
  13. Imaging our results
  14. The Cascade results
  15. Imaging the noisy simulation and QPU's output

Figures (5)

  • Figure 1: Remeshing from a 2-qubit 2D-grid into a 4-qubit 2D-grid. Each 2-qubit state expands into a structured patch of finer resolution made of $2^2=$ 4 states.
  • Figure 2: Circuit of the remeshing cascade strategy using previous results $A(\theta_{\text{opt}})$ and superposition expansion via Hadamard gates and swaps.
  • Figure 3: Comparison between the target and the quantum solution for a 15-qubit system.
  • Figure 4: 3D Heatmap of the 15 qubits noise-free simulation using the Cascade warm-start
  • Figure 5: Comparison between the classical target and quantum solutions for a 6-qubit system using Lánczos interpolation.