Learning Hamiltonians for O(1) Oracle-Query Quantum State Preparation

We propose a Hamiltonian-based quantum state preparation method implemented via a shallow parametrized quantum circuit. The approach learns the parameters of a diagonal Hamiltonian through a classical training phase, while the quantum circuit itself performs only fixed-depth Hamiltonian evolution and mixing operations. With oracle access to the learned Hamiltonian parameters, N classical data values can be encoded into n=log₂{N} qubits using O(1) quantum queries, shifting the overall computational cost to an O\(Nlog{N}\) classical preprocessing stage. For structured datasets generated by an underlying function, oracle access can be avoided by expressing the Hamiltonian in the Walsh basis and retaining only a polynomial number of significant terms. In this regime, quantum state preparation is achieved in poly(n) time using poly(n) parameters, reaching infidelities on the order of 10^-5. By restricting the Hamiltonian to one-local and two-local terms, the method naturally yields hardware-efficient circuits suitable for near-term quantum devices.