This work proves that, for sufficiently large circuit width, the nonlinear gradient flow is quantitatively approximated by a linearized NTK model, with explicit bounds depending on the number of qubits, circuit depth, circuit light-cone geometry, and the dimension of the domain of the solution to the PDE.
Abstract
We study the gradient-flow training dynamics of quantum physics-informed neural networks (QPINNs) for the solution of second-order elliptic partial differential equations with Dirichlet boundary conditions. We consider parameterized quantum circuits as function approximators and analyze their overparameterized regime through the lens of the neural tangent kernel (NTK). Our contribution is a nonasymptotic lazy-training theory for QPINNs and their variational formulation: we prove that, for sufficiently large circuit width, the nonlinear gradient flow is quantitatively approximated by a linearized NTK model, with explicit bounds depending on the number of qubits, circuit depth, circuit light-cone geometry, and the dimension of the domain of the solution to the PDE.
A unified embedding framework is introduced that formulates embedding as a functional transformation shaping the feature representation available to the variational circuit, and two embeddings within it are proposed.
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To test the potential of so-called quantum physics-informed neural network technology for solving evolutionary partial differential equations, the problem of multiple wave propagation in a weakly nonlinear medium described by the Korteweg-de Vries equation with periodic boundary conditions is considered.
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Recent advances in neural quantum states application to condensed-matter systems are reviewed, focusing on frustrated quantum magnets, interacting lattice fermions, and non-equilibrium dynamics.
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