Quantum Error Mitigation with Attention Graph Transformers for Burgers Equation Solvers on NISQ Hardware

TL;DR

The paper addresses solving the 1D viscous Burgers equation on NISQ hardware by transforming it to a linear diffusion problem via the Cole–Hopf transform, then simulating the diffusion quantum mechanically with a Trotterized circuit. A large parameter sweep generates a structured dataset used to train an attention-based graph neural network that acts as a learned error corrector for noisy quantum outputs, outperforming zero-noise extrapolation in many regimes. Across simulated and IBM hardware data, the learned QEM significantly reduces $L^2$ error while preserving shock structure and dissipation trends, enabling robust quantum CFD on near-term devices. This work demonstrates a practical, scalable hybrid quantum–classical workflow that combines physics-based solvers with data-driven error mitigation, with potential extensions to higher dimensions and broader quantum PDE solvers on NISQ hardware.

Abstract

We present a hybrid quantum-classical framework augmented with learned error mitigation for solving the viscous Burgers equation on noisy intermediate-scale quantum (NISQ) hardware. Using the Cole-Hopf transformation, the nonlinear Burgers equation is mapped to a diffusion equation, discretized on uniform grids, and encoded into a quantum state whose time evolution is approximated via Trotterized nearest-neighbor circuits implemented in Qiskit. Quantum simulations are executed on noisy Aer backends and IBM superconducting quantum devices and are benchmarked against high-accuracy classical solutions obtained using a Krylov-based solver applied to the corresponding discretized Hamiltonian. From measured quantum amplitudes, we reconstruct the velocity field and evaluate physical and numerical diagnostics, including the L2 error, shock location, and dissipation rate, both with and without zero-noise extrapolation (ZNE). To enable data-driven error mitigation, we construct a large parametric dataset by sweeping viscosity, time step, grid resolution, and boundary conditions, producing matched tuples of noisy, ZNE-corrected, hardware, and classical solutions together with detailed circuit metadata. Leveraging this dataset, we train an attention-based graph neural network that incorporates circuit structure, light-cone information, global circuit parameters, and noisy quantum outputs to predict error-mitigated solutions. Across a wide range of parameters, the learned model consistently reduces the discrepancy between quantum and classical solutions beyond what is achieved by ZNE alone. We discuss extensions of this approach to higher-dimensional Burgers systems and more general quantum partial differential equation solvers, highlighting learned error mitigation as a promising complement to physics-based noise reduction techniques on NISQ devices.

Figures (5)

• Figure 1: Data generation for classical/quantum simulations over classical and quantum hardwares of the 1D viscous Burger's equation.
• Figure 2: Quantum Error Mitigation (QEM) Graph Neural Network model schematic for training the noisy spatiotemporal circuits of 1D Burger's equation.
• Figure 3: Training and validation loss for $N=16,32,64$.
• Figure 4: QAGT-MLP vs quantum simulation results over classical computer for larger $\nu$ values $N=16,32,64$.
• Figure 5: QAGT-MLP vs quantum hardware results over quantum computer for larger $\nu$ values $N=16,32,64$.