Optimal certification of constant-local Hamiltonians

TL;DR

This paper presents an optimal protocol for certifying local Hamiltonians using only forward real-time dynamics, achieving Heisenberg-limit scaling without inverse evolution or controlled operations.

Contribution

It introduces the first intolerant Hamiltonian certification protocol that is optimal for all constant-locality Hamiltonians, using only forward dynamics.

Findings

Achieves $oxed{O(1/ ext{epsilon})}$ total evolution time for $O(1)$-local Hamiltonians.

Matches the $oxed{ ext{Omega}(1/ ext{epsilon})}$ lower bounds, reaching Heisenberg-limit scaling.

Requires no inverse evolution or controlled operations, only forward real-time dynamics.

Abstract

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 $H_0$, the goal is to decide whether $H$ is exactly equal to $H_0$ or differs from $H_0$ 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 $\Theta(1/\varepsilon)$, 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.