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arXiv:2607.29382

Algebraic Speedups for Exact Inversion of Hamiltonian Evolutions

Jizhe Lai, Mingrui Jing, Erdong Huang, Xin Wang

quant-phSubmitted Jul 31, 2026• Announced Aug 3, 2026• 0 citations• Revised Aug 4, 2026 (v2)
v1 · Jul 31, 2026v2 · Aug 4, 2026
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Abstract

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.

Comments
34 pages, 2 figures. No textual changes to the main manuscript. The Supplemental Material previously uploaded as a separate ancillary file has been merged into the main PDF for readers' convenience

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Deterministic exact inversion of an arbitrary ddd-dimensional unitary requires {Θ(d2)Θ(d^2)Θ(d2)} coherent forward calls in the worst case. We ask how this cost changes for Hamiltonian evolution U(x)=exp⁡(i∑jxjHj)U(x)=\exp(i\sum_j x_jH_j)U(x)=exp(i∑j​xj​Hj​) 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.
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