Solving Transport Equations on Quantum Computers - Potential and Limitations of Physics-Informed Quantum Circuits

Abstract

Quantum circuits with trainable parameters, paired with classical optimization routines can be used as machine learning models. The recently popularized Physics-Informed Neural Network (PINN) approach is a machine learning algorithm which solves differential equations by incorporating them into a loss function. Being a mesh-free method, it is a promising approach for computational fluid dynamics. The question arises whether the properties of quantum circuits can be leveraged for a quantum physics-informed machine learning model. In this study we compare the classical PINN-ansatz and its quantum analog, which we name the physics-informed quantum circuit (PIQC). The PIQC simulations are performed on a noise-free simulator. Studying various differential equations, we compare expressivity, accuracy and convergence properties. We find that one-dimensional problems, such as the linear transport of a Gaussian-pulse or Burgers' equation, allow a successful approximation with the classical and the quantum ansatz. For these examples, the PIQC overall performs similarly to PINN and converges more consistently and for Burgers' equations even faster. While this is promising, the chosen quantum circuit approach struggles to approximate discontinuous solutions which the classical PINN-Ansatz can represent. Based on this comparison, we extrapolate that required number of qubits for solving two-dimensional problems in aerodynamics may already be available in the next few years. However, the acceleration potential is currently unclear and challenges like noisy-circuits and approximations of discontinuous solutions have to be overcome.