LAZY TRAINING OF QUANTUM PHYSICS INFORMED NEURAL NETWORKS

This paper investigates the training dynamics of quantum physics-informed neural networks (QPINNs) used to solve second-order elliptic partial differential equations (PDEs). Specifically, it seeks to determine whether these quantum models, like their classical counterparts, operate in a lazy-training regime—a state where the network parameters remain close to their initial values, allowing the training process to be described by a linearized model known as the Neural Tangent Kernel (NTK).

The authors analyze the gradient-flow training dynamics of QPINNs through the lens of the NTK. They derive nonasymptotic bounds for the discrepancy between the actual nonlinear training dynamics and the linearized NTK model. The study covers two formulations: one based on the differential operator of the PDE and a variational formulation. The authors provide explicit bounds that account for the number of qubits, circuit depth, light-cone geometry, and the dimension of the spatial domain.

The study establishes that for sufficiently large circuit width, the training of QPINNs is quantitatively approximated by a linearized NTK model. A significant finding is that the variational formulation of the QPINN requires weaker growth conditions on the circuit depth (scaling as L^5) compared to the standard differential operator approach (scaling as L^9). Furthermore, the variational approach is shown to be more robust, as it does not require restrictive constraints on the PDE coefficients, allowing for a broader class of elliptic problems to be addressed.

This work provides a rigorous mathematical foundation for using quantum neural networks in scientific computing. By proving that QPINNs can be effectively analyzed via the NTK framework, the authors offer a path to understanding the convergence and stability of these models. The results help researchers predict how quantum circuits will behave during training, which is crucial for developing reliable quantum algorithms for solving complex physical systems.