Using Lemma 6 and Lemma 7, we have ‖𝑊𝑗‖𝐹 ≤𝜋‖𝐼−𝐶 𝑗‖𝐹 ≤𝜋𝑇‖𝐻−𝐻 𝑗‖𝐹 ≤2𝜋𝑇 √𝑚 𝜀,(D1) and ‖𝑊𝑗‖∞ ≤𝜋‖𝐼−𝐶 𝑗‖∞ ≤𝜋𝑇‖𝐻−𝐻 𝑗‖∞ ≤2𝜋𝑇 𝑚𝜀,(D2) since‖𝐻−𝐻 𝑗‖ℓ∞ ≤𝜀and both𝐻, 𝐻 𝑗 are𝑚-sparse

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• Heisenberg-limited Hamiltonian learning without short-time control quant-ph · 2026-04-30 · unverdicted · none · ref 63

  Heisenberg-limited Hamiltonian learning is achievable with any constant minimum evolution time T per query, attaining optimal 1/ε total-time scaling for logarithmically sparse Hamiltonians.