The power of time reversal in Hamiltonian property testing
Abstract
Quantum algorithms enable Heisenberg-limited scaling across many tasks in sensing and learning, achieving precision ε using O(1/ε) total evolution time. However, in Hamiltonian property testing, the best known algorithms have O(1/ε2) total evolution time, unless it is possible to reverse the flow of time under the Hamiltonian. In this work, we prove that time reversal is necessary to close this gap. We establish evolution time lower bounds for Hamiltonian certification and locality testing, for a continuous family of error metrics ranging from average- to worst-case. We also prove matching upper bounds for Hamiltonian certification in all cases, both with and without time reversal, showing that the advantage from time reversal persists. Our results follow from a new Fourier-analytic approach to quantum lower bounds in continuous time, which is sensitive to time reversal and applicable beyond Hamiltonian property testing. This gives an alternative proof that inverses are required for amplitude amplification and estimation, improving the lower bounds of Tang and Wright and extending to fractional queries. We also show that, for unstructured search, uncertainty in the phase of the oracle destroys the quantum speedup without inverse queries, even when the oracle is fixed and coherent. Our work rigorously establishes time reversal as a resource governing both the precision of quantum measurement and the robustness of quantum algorithms.
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BibTeX
@misc{allen2026power,
title = {{The power of time reversal in Hamiltonian property testing}},
author = {Richard R. Allen and Matthias C. Caro and Yanlin Chen and Tom Gur and Angus Lowe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04996},
eprint = {2610.04996},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04996},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The power of time reversal in Hamiltonian property testing
AU - Richard R. Allen
AU - Matthias C. Caro
AU - Yanlin Chen
AU - Tom Gur
AU - Angus Lowe
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04996
UR - https://arxiv.org/abs/2610.04996
AB - Quantum algorithms enable Heisenberg-limited scaling across many tasks in sensing and learning, achieving precision ε using O(1/ε) total evolution time. However, in Hamiltonian property testing, the best known algorithms have O(1/ε^2) total evolution time, unless it is possible to reverse the flow of time under the Hamiltonian. In this work, we prove that time reversal is necessary to close this gap. We establish evolution time lower bounds for Hamiltonian certification and locality testing, for a continuous family of error metrics ranging from average- to worst-case. We also prove matching upper bounds for Hamiltonian certification in all cases, both with and without time reversal, showing that the advantage from time reversal persists. Our results follow from a new Fourier-analytic approach to quantum lower bounds in continuous time, which is sensitive to time reversal and applicable beyond Hamiltonian property testing. This gives an alternative proof that inverses are required for amplitude amplification and estimation, improving the lower bounds of Tang and Wright and extending to fractional queries. We also show that, for unstructured search, uncertainty in the phase of the oracle destroys the quantum speedup without inverse queries, even when the oracle is fixed and coherent. Our work rigorously establishes time reversal as a resource governing both the precision of quantum measurement and the robustness of quantum algorithms.
ER -