Quantum Hamiltonian Certification

We formalize and study the Hamiltonian certification problem, a fundamental task in quantum physics, crucial for verifying the accuracy of quantum simulations and quantum-enhanced technologies. Given access to e -iHt for an unknown Hamiltonian H, the goal of the problem is to determine whether H is ε 1 -close to or ε 2 -far from a target Hamiltonian H 0 . While Hamiltonian learning methods have been extensively studied, they often require restrictive assumptions and suffer from inefficiencies when adapted for certification tasks.

This work introduces a direct and efficient framework for Hamiltonian certification, which distinguishes whether an unknown Hamiltonian matches a target specification within given precision bounds. Our approach achieves optimal total evolution time Θ((ε 2 -ε 1 ) -1 ) for certification under the normalized Frobenius norm, without prior structural assumptions. This approach also extends to certify Hamiltonians with respect to all Pauli norms and normalized Schatten p-norms for 1 ≤ p ≤ 2 in the one-sided error setting (ε 1 = 0), where the optimality is consistently maintained. Notably, the result in Pauli 1-norm suggests a quadratic advantage of our approach over all possible Hamiltonian learning approaches. We also establish matching lower bounds to show the optimality of our approach across all the above settings. We complement our result by showing that the certification problem with respect to normalized Schatten ∞-norm is coQMA-hard, and therefore unlikely to have efficient solutions. This hardness result provides strong evidence that our focus on above metrics is not merely a technical choice but a requirement for efficient certification.

To enhance practical applicability, we develop an ancilla-free certification method that maintains the inverse precision scaling while eliminating the need for auxiliary qubits, making our approach immediately accessible for near-term quantum devices with limited resources.