We develop an exact spectral-response theory for the Grover-type iterate $W_T=D_{\xi }exp(-\mathrm{i}TH)$, in which the evolution generated by a diagonal Ising Hamiltonian is used directly as a continuous phase oracle. An energy-grouped recurrence and its generating-function solution show how the empirical characteristic function determines the position, width, height, and saturation time of an energy-selective resonance. For an annealed Gaussian density of states, a high-density-tail resonance containing $M$ configurations is reached after $\mathrm{\Theta }\left(\sqrt{2^n/M}\right)$ oracle calls with success probability $\mathrm{\Theta }\left(1\right)$, giving a quadratic query improvement over independent uniform sampling with classical energy evaluation. For correlated random Ising spectra, overlap-dependent covariances lead to a realization-dependent resonance shift with root-mean-square scale $O\left(n^{-3}\right)$, parametrically larger than the resonance width, and can also reduce the peak height. The shift is both an algorithmic detuning and a coherent probe of sample-specific spectral fluctuations whose ensemble statistics reflect the Ising overlap structure. Spectral symmetrization and iterative calibration can remove or compensate the resonance-center displacement for prescribed-energy targeting. We also clarify the relation to designed spectral filters and the precision and coherence requirements of this asymptotic primitive.

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• Optimization problems
• Quantum algorithms
• Quantum computation
• Disordered systems
• Ising model