Optimal certification of constant-local Hamiltonians
Junseo Lee, Myeongjin Shin
TL;DR
The paper proves that intolerant certification of constant-locality Hamiltonians can be achieved with total evolution time scaling as Θ(1/ε), matching lower bounds and attaining Heisenberg-limited precision, using only forward real-time dynamics. It introduces Bell-sampling as a spectral probe and a gap-statistic Λ(H,ε) to link eigenvalue structure to detectability of deviations from a target Hamiltonian. The approach combines randomized diagonal-basis selection, Pauli twirling to project onto an effective diagonal, stability analysis of eigenvalue gaps under perturbations, and Trotterized implementations to realize the required evolutions, yielding a general algorithm with time bound O(c^k/ε) for any k-local Hamiltonian and Θ(1/ε) for O(1)-local cases. This work advances certification without inverse or controlled evolutions and clarifies the role of locality in achieving optimal certification performance, while outlining clear open directions for tolerant certification, locality scaling, and extensions to broader dynamical models.
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 $Θ(1/\varepsilon)$, 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.