Multi-parameter function estimation in general Hamiltonians

Estimating extensive parameters encoded in a Hamiltonian is a central task in quantum sensing and learning. While the ultimate precision limit for estimating a single parameter coupled to a single generator is well established, the corresponding bound for estimating functions of multiple parameters—each coupled to distinct and possibly noncommuting generators—remains unknown. Here, we derive the ultimate limit and an optimal estimation protocol for any linear function of parameters in a general Hamiltonian. Remarkably, we show that despite the inherently multi-parameter nature of the problem, the optimal bound can be expressed in terms of the single-parameter quantum Cramér–Rao bound and can be saturated. Our result unifies and extends previous works, which were restricted to commuting generators, providing a general framework for optimal function estimation in quantum systems with arbitrary generator structure.