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\(c^k/varepsilon\) 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 Θ\(1/varepsilon\), matching the known Ω\(1/varepsilon\) 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.