Hamiltonian locality testing and certification do not achieve the Heisenberg limit
Abstract
We establish lower bounds for Hamiltonian property testing with access to the time-evolution operator but not its inverse. Each experiment may query the time-evolution operator multiple times, and distances between Hamiltonians are measured in the normalized Frobenius norm. In this model, we show that testing whether a Hamiltonian is k-local or ε-far from every k-local Hamiltonian requires Ω(1/ε2) total evolution time, matching the upper bound of Kallaugher and Liang (TQC'25). We also prove that testing whether an unknown Hamiltonian equals a target Hamiltonian or is ε-far from it requires Ω(1/ε2) total evolution time, matching the upper bound of Sinha and Tong (2025). These are the first lower bounds for natural problems in Hamiltonian learning and testing that rule out Heisenberg-limited scaling of 1/ε. As a third result, we show that amplitude estimation to precision ε requires Ω(1/ε2) total time evolution, recovering the result of Tang and Wright (QIP'26) in the continuous-time query model. All three results follow from the hardness of distinguishing the zero Hamiltonian from a suitably chosen ensemble of random Hamiltonians. We establish this hardness by adapting the continuous-time adversary method to forward Hamiltonian evolution.
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BibTeX
@misc{gutierrez2026hamiltonian,
title = {{Hamiltonian locality testing and certification do not achieve the Heisenberg limit}},
author = {Francisco Escudero Gutiérrez and Junseo Lee and Sebastian Zur},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03205},
eprint = {2610.03205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03205},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hamiltonian locality testing and certification do not achieve the Heisenberg limit
AU - Francisco Escudero Gutiérrez
AU - Junseo Lee
AU - Sebastian Zur
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.03205
UR - https://arxiv.org/abs/2610.03205
AB - We establish lower bounds for Hamiltonian property testing with access to the time-evolution operator but not its inverse. Each experiment may query the time-evolution operator multiple times, and distances between Hamiltonians are measured in the normalized Frobenius norm. In this model, we show that testing whether a Hamiltonian is k-local or ε-far from every k-local Hamiltonian requires Ω(1/ε^2) total evolution time, matching the upper bound of Kallaugher and Liang (TQC'25). We also prove that testing whether an unknown Hamiltonian equals a target Hamiltonian or is ε-far from it requires Ω(1/ε^2) total evolution time, matching the upper bound of Sinha and Tong (2025). These are the first lower bounds for natural problems in Hamiltonian learning and testing that rule out Heisenberg-limited scaling of 1/ε. As a third result, we show that amplitude estimation to precision ε requires Ω(1/ε^2) total time evolution, recovering the result of Tang and Wright (QIP'26) in the continuous-time query model. All three results follow from the hardness of distinguishing the zero Hamiltonian from a suitably chosen ensemble of random Hamiltonians. We establish this hardness by adapting the continuous-time adversary method to forward Hamiltonian evolution.
ER -