Bridging quantum and classical computing for partial differential equations through multifidelity machine learning

TL;DR

The paper tackles the hardware bottlenecks of near-term quantum PDE solvers by introducing a multifidelity learning framework that fuses abundant low-fidelity quantum data with sparse high-fidelity classical data. It employs Kolmogorov-Arnold networks to separately learn linear and nonlinear correction mappings and blends them with a learnable parameter to produce high-fidelity predictions beyond the quantum training window. Demonstrations on viscous Burgers and 2D lid-driven cavity flows show substantial error reductions and robust extrapolation, validating a practical pathway for quantum-assisted scientific computing on current hardware. The work highlights how to reduce reliance on expensive high-fidelity simulations, enabling immediate value from quantum devices while providing a scalable route for future quantum PDE algorithm deployment.

Abstract

Quantum algorithms for partial differential equations (PDEs) face severe practical constraints on near-term hardware: limited qubit counts restrict spatial resolution to coarse grids, while circuit depth limitations prevent accurate long-time integration. These hardware bottlenecks confine quantum PDE solvers to low-fidelity regimes despite their theoretical potential for computational speedup. We introduce a multifidelity learning framework that corrects coarse quantum solutions to high-fidelity accuracy using sparse classical training data, facilitating the path toward practical quantum utility for scientific computing. The approach trains a low-fidelity surrogate on abundant quantum solver outputs, then learns correction mappings through a multifidelity neural architecture that balances linear and nonlinear transformations. Demonstrated on benchmark nonlinear PDEs including viscous Burgers equation and incompressible Navier-Stokes flows via quantum lattice Boltzmann methods, the framework successfully corrects coarse quantum predictions and achieves temporal extrapolation well beyond the classical training window. This strategy illustrates how one can reduce expensive high-fidelity simulation requirements while producing predictions that are competitive with classical accuracy. By bridging the gap between hardware-limited quantum simulations and application requirements, this work establishes a pathway for extracting computational value from current quantum devices in real-world scientific applications, advancing both algorithm development and practical deployment of near-term quantum computing for computational physics.

Figures (12)

• Figure 1: Graphical abstract: overview of the hybrid quantum-classical multifidelity framework. The quantum solver (QLBM circuit, bottom left) generates abundant low-fidelity data $\mathbf{q}^{\text{quantum}}$ used to train the low-fidelity network $\mathcal{K}_{\text{LF}}$. The classical solver (bottom right) provides sparse high-fidelity data $\mathbf{q}^{\text{classical}}$ for training the correction networks. The nonlinear network $\mathcal{K}_{\text{nl}}$ and linear network $\mathcal{K}_{\text{lin}}$ take the low-fidelity predictions as input and are blended via the learned parameter $\alpha$ to produce the final multifidelity prediction $\mathbf{q}_{\text{MF}}$.
• Figure 2: Schematic quantum circuit for a single 1D Burgers QLBM step. The ancilla-based block encoding (LCU) implements the non-unitary collision operator via controlled diagonals $V_1$ and $V_2$, and the streaming operator $S$ acts on the lattice register $|lat\rangle$ controlled by the link register $|link\rangle$, realizing the shift $x_j \mapsto x_j \pm 1$. A final Hadamard on $|link\rangle$ represents summation over discrete velocities for macroscopic retrieval; in this work the circuit is classically emulated via statevector simulation rather than hardware measurements.
• Figure 3: Vorticity circuit for the D2Q5 QLBM-frugal scheme lee2024frugal. The ancilla qubit $|0\rangle_a$ controls block encoding of the non-unitary collision operator via diagonal gates $C_1$ (controlled on $|0\rangle$) and $C_2$ (controlled on $|1\rangle$). The streaming operator $S$ shifts the lattice register $|lat\rangle$ controlled by the link register $|link\rangle$. A final Hadamard layer on $|link\rangle$ sums over velocity directions for macroscopic retrieval. In the implementation, the circuit uses 12 qubits total (8 spatial + 3 link + 1 ancilla), and vorticity boundary conditions are imposed classically after each quantum update.
• Figure 4: Stream function circuit for the D2Q5 QLBM-frugal scheme lee2024frugal. The structure mirrors the vorticity circuit: block-encoded collision via $C_1$ and $C_2$ controlled by ancilla $|0\rangle_a$, streaming $S$ controlled by $|link\rangle$, and Hadamard summation on $|link\rangle$ for macroscopic retrieval. The vorticity source term $S=-\omega$ is embedded directly into the input state amplitudes rather than via a separate source register. The circuit uses 13 qubits total (8 spatial + 3 link + 2 ancilla), and $\psi=0$ boundary conditions are imposed classically after each quantum update.
• Figure 5: Training loss evolution for the Burgers equation (B4). The vertical dashed line marks the transition from LF to MF training.