Solving the 3D Heat Equation with VQA Via Remeshing-Based Warm Starts Demonstrates Potential for Solving Classically Intractable Problems

Solving complex equations that describe how heat flows through materials presents a significant challenge for modern computing, yet holds the key to advances in many engineering fields. Samuel Donachie, Ulysse Remond, Arthur Mathorel, and Kyryl Kazymyrenko from EDF R and D and Sorbonne University now demonstrate a promising new approach using variational quantum algorithms, a type of hybrid quantum-classical computation suited to current, imperfect quantum hardware. The team tackles this problem by formulating the heat transfer equation as a system of linear equations and then optimises a cost function representing the system’s thermal energy, employing a novel remeshing strategy. This method intelligently refines the computational mesh while reusing previously optimised parameters, effectively bypassing common challenges in quantum computation and achieving convergence with increasing resolution, offering a practical pathway towards applying quantum algorithms to real-world engineering problems.

Cascade Warm Start for Variational Quantum Solvers

This research investigates the application of Variational Quantum Algorithms (VQAs) to solve a complex 3D heat equation, exploring how well quantum computers can represent and solve challenging physical problems. The team developed a remeshing-based warm-start method, called Cascade, which iteratively refines a mesh representing the 3D space and uses the solution from a coarser mesh as a starting point for a finer mesh, significantly improving the optimization process for the VQC. Researchers employed parameterized quantum circuits (PQCs) with two main designs, Ry-CNOT and Rx-Ry-CNOT, and conducted extensive simulations, including those incorporating realistic noise, as well as execution on actual quantum hardware, specifically the IonQ Aria 1. Results demonstrate that the Cascade method improves the optimization of the VQC, leading to more accurate solutions.

The Rx-Ry-CNOT design generally performed better than the Ry-CNOT design, likely because fewer entangling operations reduce the impact of errors. As the number of qubits increased, the accuracy of the solutions decreased due to the inherent randomness in quantum measurements, requiring more measurements to achieve statistically reliable results. The introduction of noise, through both simulation and the IonQ Aria 1, degraded accuracy, but the general shape and gradients of the solutions were preserved, indicating the chosen design effectively captures the essential features of the problem. The results from the IonQ Aria 1 closely matched the noisy simulations, validating the accuracy of the noise model.

This research proposes a new benchmarking strategy for quantum hardware, focusing on evaluating the hardware’s ability to reproduce a known quantum state rather than attempting the full optimization process. The team used parameters obtained from classical optimization, using the Cascade method, to generate a target quantum state and measured the accuracy of the hardware’s state reproduction using energy and fidelity metrics, comparing the performance to noise models. This research offers a practical benchmarking strategy for evaluating quantum hardware and helps validate the accuracy of noise models essential for simulating quantum algorithms. The findings provide insights into the effectiveness of different designs and optimization techniques for solving complex physical problems and highlight the challenges of scaling quantum algorithms to larger qubit counts. This research presents a promising approach to solving complex physical problems using VQCs, along with a novel benchmarking strategy for evaluating quantum hardware, paving the way for future advancements in the field.

Variational Quantum Algorithms Solve 3D Heat Transfer

This work presents a methodology for solving complex 3D heat equations using Variational Quantum Algorithms (VQAs), demonstrating a practical approach to leveraging quantum computing for challenging scientific problems. Researchers defined a cost function that encodes the thermal energy of a system and minimizes it using a parameterized quantum state, ultimately yielding the solution to the heat equation. The team successfully formulated the problem as a variational equation, enabling the use of hybrid quantum-classical algorithms suitable for current Noisy Intermediate-Scale Quantum (NISQ) devices. A key achievement is the introduction of a remeshing strategy designed to improve the trainability and convergence of VQAs.

This strategy progressively increases the resolution of the discretization by reusing optimized parameters from coarser representations, creating tailored warm starts for the quantum algorithm. The optimization process centers on minimizing the cost function, which is directly linked to the thermal energy of the system. Researchers analytically minimized the variational equation with respect to a normalization factor, resulting in an expression crucial for accurately representing the solution. Experiments demonstrate convergence of scalar quantities with mesh refinement, confirming the effectiveness of the remeshing approach.

This work establishes a concrete, structured target for quantum hardware by focusing on reproducing physically meaningful states corresponding to the solution of the 3D heat equation. The team’s approach offers a valuable and application-driven benchmark for evaluating the capabilities of future quantum devices. This research demonstrates a methodology for applying Variational Quantum Algorithms to solve partial differential equations, specifically the 3D heat equation discretized using the finite element method. Researchers successfully minimized a cost function representing thermal energy by employing a noise-free simulation of a VQA, addressing the challenges of optimization as system size increases.

A key achievement is the introduction of the Cascade protocol, a remeshing-based warm-start strategy that reuses optimized parameters from coarser discretizations to initialize larger systems, enabling convergence with a 15-qubit system. Extensive simulations benchmarked various designs and initialization strategies, revealing that more expressive circuits gradually outperformed simpler configurations as system size increased. The Cascade protocol significantly improved convergence stability and accuracy at larger scales, exceeding both cold and uniform start baselines, and offering a scalable, physically motivated approach to training VQAs for PDEs. While acknowledging the limitations of current quantum hardware, the authors propose a structured benchmark for evaluating quantum processing units by comparing noise-free simulations, noisy models, and outputs from real devices.

Fidelity and energy-based accuracy metrics suggest that small systems can already be meaningfully benchmarked using this approach. Future research directions include exploring problem-specific warm starts, new optimization algorithms, and hardware-aware design. This methodology may ultimately serve as a foundation for solving real-world PDEs and assessing quantum advantage in scientific computing as quantum hardware continues to mature.