Precision limits for time-dependent quantum metrology under Markovian noise

We derive ultimate precision bounds for estimating parameters encoded in time-dependent Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields Q_coh(T)sim T^2k, then under Markovian noise the QFI scales at most as Q(T)sim T^2k in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to Q(T)sim T^2k-1. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting T⁴ and T³ scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions - supplemented by spin-squeezed probes - that asymptotically saturate the bounds, establishing their tightness.