Optimal certification of constant-local Hamiltonians

We study the problem of certifying local Hamiltonians from real-time access to their dynamics. Given oracle access to e-itH for an unknown k-local Hamiltonian H and a fully specified target Hamiltonian H0, the goal is to decide whether H is exactly equal to H0 or differs from H0 by at least varepsilon in normalized Frobenius norm, while minimizing the total evolution time. We introduce the first intolerant Hamiltonian certification protocol that achieves optimal performance for all constant-locality Hamiltonians. For general n-qubit, k-local, traceless Hamiltonians, our procedure uses O(ck/varepsilon) total evolution time for a universal constant c, and succeeds with high probability. In particular, for O(1)-local Hamiltonians, the total evolution time becomes Theta(1/varepsilon), matching the known Omega(1/varepsilon) lower bounds and achieving the gold-standard Heisenberg-limit scaling. Prior certification methods either relied on implementing inverse evolution of H, required controlled access to e-itH, or achieved near-optimal guarantees only in restricted settings such as the Ising case (k=2). In contrast, our algorithm requires neither inverse evolution nor controlled operations: it uses only forward real-time dynamics and achieves optimal intolerant certification for all constant-locality Hamiltonians.