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on System-size lower bounds for Hamiltonian learning

The statement

For learning an $n$ -qubit local Hamiltonian from real-time evolution to coefficient error $\varepsilon$ , must the total evolution time depend on $n$ ? Can one prove a lower bound of order $\log(n)/\varepsilon$ , beyond the $\Omega(1/\varepsilon)$ bound stated in the 2024 source? Is dependence on the one-spin energy $\max_{i\in[n]}\sum_{a:\,i\in\operatorname{supp}(E_a)}\|\lambda_aE_a\|$ necessary? See the preceding source slide for the learning model.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.