Interpolation of unitaries with time-dependent Hamiltonians via Deep Learning

Quantum systems governed by time-dependent Hamiltonians pose significant challenges for the accurate computation of unitary time-evolution operators, which are essential for predicting quantum state dynamics. In this work, we introduce a physics-informed deep learning approach based on Physics-Informed Neural Networks to estimate these operators over the full time domain. By incorporating physical constraints such as unitarity and leveraging the second-order Magnus expansion on the evolution operator, the proposed framework enables the estimation of unitary matrices at different time intervals. The model is trained using simulated unitary operators and evaluated on quantum systems ranging from 2 to 6 qubits. For larger many-body systems, specifically those with 7 and 8 qubits, the same methodology is employed to reconstruct an effective time-dependent Hamiltonian, from which the corresponding time-evolution operator is computed over the entire temporal domain. The proposed framework achieves fidelities exceeding 0.92 using a limited number of unitary samples, indicating a potential reduction in measurement and data acquisition costs. These results highlight the effectiveness of the approach for data-driven simulation and identification of quantum dynamical systems, with direct relevance to quantum computing and quantum simulation applications.