哈密顿演化的精确逆变换的代数加速
Algebraic Speedups for Exact Inversion of Hamiltonian Evolutions
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中文总结 AI 辅助
该研究针对参数隐藏的哈密顿演化精确逆变换，利用本征值加性关系与对称扇区特性建立查询复杂度界，为相关量子协议的未知动力学逆转提供了结构依赖的高效方法。
中文摘要 AI 辅助
对任意d维幺正矩阵的确定性精确逆变换，最坏情况需要Θ(d²)次相干正向调用。我们研究当生成元已知但参数隐藏时，哈密顿演化U(x)=exp(i∑ⱼxⱼHⱼ)的该成本如何变化。对于具有固定本征基的单参数族，我们证明不同本征值间的加性关系恰好决定最优查询次数，并构造相应的逆变换协议。对于一般族，我们证明重复对称扇区不影响精确查询复杂度，且给出将不等价有效扇区逆变换组合的自动构造。我们还给出充分相位对齐条件，在此条件下，族特定结构可减少查询次数。这些结果为逆转Tavis-Cummings时序外关联器协议、集体自旋回波验证及被动多模链路中产生的未知动力学建立了依赖结构的界，无需预先了解或显式估计底层耦合强度。
英文摘要
Deterministic exact inversion of an arbitrary $d$-dimensional unitary requires {$Θ(d^2)$} coherent forward calls in the worst case. We ask how this cost changes for Hamiltonian evolution $U(x)=\exp(i\sum_j x_jH_j)$ when the generators are known but the parameters are hidden. For one-parameter families with a fixed eigenbasis, we show that additive relations among the distinct eigenvalues determine the optimal query number exactly, and we construct the corresponding inversion protocol. For general families, we prove that repeated symmetry sectors do not affect the exact query complexity and give an automatic construction for combining inverses from inequivalent active sectors. We also give a sufficient phase-alignment condition under which family-specific structure can reduce the query number. These results establish structure-dependent bounds for reversing the unknown dynamics arising in Tavis-Cummings out-of-time-order correlator protocols, collective-spin echo verification, and passive multimode links, without requiring prior knowledge or explicit estimation of the underlying coupling strengths.
