基于伊辛哈密顿量相位或acles的能量选择性量子搜索
Spectral theory of energy-selective quantum search with Ising Hamiltonian phase oracles
AI总结：
研究使用伊辛哈密顿量作为相位或acles的能量选择性量子搜索原语，通过交替与Grover扩散算子产生Grover型放大峰，并给出峰值位置、宽度和高度的精确谱递推与生成函数表示。
摘要：
伊辛哈密顿量是无序磁体的基本模型，也是量子与经典优化的标准语言。我们研究了一种能量选择性量子搜索原语，其中物理演化\(\exp(-\mathrm{i} T H)\)直接用作哈密顿量相位或acles。与布尔或acles不同，该或acles通过相位连续标记配置，并选择有限共振带而非预设标记集。我们证明，将其与Grover扩散算子交替使用仍能产生Grover型放大峰。精确的谱递推和生成函数表示确定了峰值位置、宽度和高度。对于退火高斯态密度，当共振包含\(M\)个配置时，高密度尾部的目标能量需要\(\Theta(\sqrt{2^n/M})\)次或acles调用。对于随机伊辛谱，重叠诱导的关联会偏移和扭曲峰值；谱对称化和迭代校准可消除针对指定能量目标的这种失谐。
英文摘要：
We develop an exact spectral-response theory for the Grover-type iterate \(W_T=D_ξ\exp(-\mathrm{i} T H)\), 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 \(Θ(\sqrt{2^n/M})\) 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.
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