mlsanityData to 14 Oct 2026

Hamiltonian locality testing and certification do not achieve the Heisenberg limit

arXiv 2610.03205 ↗First posted 2 Oct 2026quant-ph cs.CC cs.DScited by 0

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Abstract (arXiv)

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 varepsilon-far from every k-local Hamiltonian requires Omega(1/varepsilon2) 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 varepsilon-far from it requires Omega(1/varepsilon2) 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/varepsilon. As a third result, we show that amplitude estimation to precision varepsilon requires Omega(1/varepsilon2) 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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References · 20

  1. 2020Sample-efficient learning of quantum many-body systemscited by 37
  2. 2022Learning many-body Hamiltonians with Heisenberg-limited scalingcited by 23
  3. 2022Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrixcited by 23
  4. 2023Learning quantum Hamiltonians at any temperature in polynomial timecited by 23
  5. 2022Practical Black Box Hamiltonian Learningcited by 16
  6. 2024Structure learning of Hamiltonians from real-time evolutioncited by 9
  7. 2024Learning k-body Hamiltonians via compressed sensingcited by 8
  8. 2024Hamiltonian Property Testingcited by 6
  9. 2025Amplitude amplification and estimation require inversescited by 6
  10. 2024Learning the structure of any Hamiltonian from minimal assumptionscited by 5
  11. 2025Nearly optimal algorithms to learn sparse quantum Hamiltonians in physically motivated distancescited by 5
  12. 2024Testing and learning structured quantum Hamiltonianscited by 4
8 more
  1. 2025Ansatz-free Hamiltonian learning with Heisenberg-limited scalingcited by 4
  2. 2025Hamiltonian Locality Testing via Trotterized Postselectioncited by 3
  3. 2026Optimal detection of dissipation in Lindbladian dynamicscited by 3
  4. 2026Heisenberg-limited Hamiltonian learning without short-time controlcited by 3
  5. 2026Optimal Ansatz-free Hamiltonian Learning In Situcited by 3
  6. 2026Certifying and learning local quantum Hamiltonianscited by 2
  7. 2026Learning the structure of open quantum systemscited by 2
  8. 2026Characterizing Arbitrary Lindbladian Dynamics with a Few Pauli Measurementscited by 1

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