Spectral theory of energy-selective quantum search with Ising Hamiltonian phase oracles
Abstract
We develop an exact spectral-response theory for the Grover-type iterate WT=Dξexp(−i T H), 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 Θ(√(2n/M)) oracle calls with success probability Θ(1), 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(n−3), 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.
Versions
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BibTeX
@misc{plyashechnik2026spectral,
title = {{Spectral theory of energy-selective quantum search with Ising Hamiltonian phase oracles}},
author = {A. S. Plyashechnik and A. A. Zhukov and A. V. Lebedev and W. V. Pogosov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.03380},
eprint = {2606.03380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.03380},
}
RIS
TY - UNPB
TI - Spectral theory of energy-selective quantum search with Ising Hamiltonian phase oracles
AU - A. S. Plyashechnik
AU - A. A. Zhukov
AU - A. V. Lebedev
AU - W. V. Pogosov
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2606.03380
UR - https://arxiv.org/abs/2606.03380
AB - We develop an exact spectral-response theory for the Grover-type iterate W_T=D_ξexp(−i T H), 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 Θ(√(2^n/M)) oracle calls with success probability Θ(1), 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(n^−3), 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.
ER -