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= 2 N qubits using O(1) quantum queries, shifting the overall computational cost to an O(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. This separation keeps the quantum execution shallow while moving optimization to classical computation.