Analysis of schemes for achieving Heisenberg-limited non-Markovian metrology

Frequency estimation concerns determining the frequency ω associated with the signal generator G of the unitary evolution U = exp(-i ω t G). The quantum Fisher information (QFI) quantifies the ultimate attainable precision of ω as a function of the interrogation time t. While the QFI scales as O(t²) for noiseless systems, this scaling typically degrades under generic non-Markovian noise without control. Modeling the noise as an interaction with an environment of possibly unbounded spectrum, with operators E_i (called the error generators) coupling the system to the environment, we introduce Zeno-limited control schemes employing fast control operations. In the limit of vanishing control intervals, δt →0, these schemes asymptotically recover O(t²) scaling when specific algebraic conditions between signal and error generators are satisfied. We study three classes of schemes: a) approximate quantum error detection, b) approximate quantum error correction, and c) deterministic dynamical decoupling and establish the conditions for their constructions. We further analyze leading order errors and validate performance through numerical simulations.