The paper addresses the problem of Hamiltonian structure learning from real-time evolution, aiming to recover an unknown local Hamiltonian \( H = \sum_{a=1}^{m} \lambda_a E_a \) acting on \( n \) qubits. The goal is to achieve this recovery with optimal Heisenberg-limited scaling in the error, i.e., \( t_{\text{total}} = \Theta(\log(n)/\epsilon) \), while also supporting Hamiltonians without bounded range and achieving constant time resolution \( t_{\text{min}} = \Theta(1/\tau) \). The main contributions of the paper include: 1. **A recursive framework for Hamiltonian learning**: The algorithm uses a bootstrapping approach to improve estimates of the Hamiltonian coefficients iteratively, reducing the error by a factor of 2 at each step. 2. **Improving estimates with continuous quantum control**: The algorithm leverages continuous quantum control to estimate the coefficients of the Hamiltonian \( H - H_0 \) to a given error, using the Trotterization technique to approximate the evolution. 3. **Constant-time Trotterization**: A novel bound on the Trotter error is derived, allowing for constant time resolution in the alternating evolution between the unknown and known Hamiltonians. 4. **Efficient structure learning**: The algorithm uses Goldreich–Levin-like queries to efficiently identify the non-zero interaction terms in the Hamiltonian, achieving fixed-parameter tractable (FPT) runtime. The paper also discusses the limitations of existing Hamiltonian learning algorithms and provides a detailed technical overview of the proposed approach, including bounds on nested commutators and the implementation of the Trotter approximation. The results are supported by theoretical guarantees and experimental validation, demonstrating the effectiveness of the proposed method in recovering the Hamiltonian structure and achieving the desired scaling in total evolution time and time resolution.The paper addresses the problem of Hamiltonian structure learning from real-time evolution, aiming to recover an unknown local Hamiltonian \( H = \sum_{a=1}^{m} \lambda_a E_a \) acting on \( n \) qubits. The goal is to achieve this recovery with optimal Heisenberg-limited scaling in the error, i.e., \( t_{\text{total}} = \Theta(\log(n)/\epsilon) \), while also supporting Hamiltonians without bounded range and achieving constant time resolution \( t_{\text{min}} = \Theta(1/\tau) \). The main contributions of the paper include: 1. **A recursive framework for Hamiltonian learning**: The algorithm uses a bootstrapping approach to improve estimates of the Hamiltonian coefficients iteratively, reducing the error by a factor of 2 at each step. 2. **Improving estimates with continuous quantum control**: The algorithm leverages continuous quantum control to estimate the coefficients of the Hamiltonian \( H - H_0 \) to a given error, using the Trotterization technique to approximate the evolution. 3. **Constant-time Trotterization**: A novel bound on the Trotter error is derived, allowing for constant time resolution in the alternating evolution between the unknown and known Hamiltonians. 4. **Efficient structure learning**: The algorithm uses Goldreich–Levin-like queries to efficiently identify the non-zero interaction terms in the Hamiltonian, achieving fixed-parameter tractable (FPT) runtime. The paper also discusses the limitations of existing Hamiltonian learning algorithms and provides a detailed technical overview of the proposed approach, including bounds on nested commutators and the implementation of the Trotter approximation. The results are supported by theoretical guarantees and experimental validation, demonstrating the effectiveness of the proposed method in recovering the Hamiltonian structure and achieving the desired scaling in total evolution time and time resolution.