Energy-selective quantum search with Ising Hamiltonian phase oracles

Ising Hamiltonians are basic models of disordered magnets and a standard language for quantum and classical optimization. We study an energy-selective quantum search primitive in which the physical evolution exp\(-i T H\) is used directly as a Hamiltonian phase oracle. Unlike a Boolean oracle, this oracle marks configurations continuously by their phases and selects a finite resonance band rather than a preassigned marked set. We show that alternating it with the Grover diffusion operator nevertheless produces a Grover-type amplification peak. An exact spectral recurrence and a generating-function representation determine the peak position, width, and height. For an annealed Gaussian density of states, target energies in a high-density tail require Θ\(sqrt{2ⁿ/M}\) oracle calls when the resonance contains M configurations. For random Ising spectra, overlap-induced correlations shift and distort the peak; spectral symmetrization and iterative calibration remove this detuning for prescribed-energy targeting.