  Quantum metrology is a rapidly developing branch of quantum technologies.
While various theories have been established on quantum metrology for Markovian
processes, i.e., quantum channel estimation, quantum metrology for
non-Markovian processes is much less explored. In this Letter, we establish a
general framework of non-Markovian quantum metrology. For any parametrized
non-Markovian process on a finite-dimensional system, we derive a formula for
the maximal amount of quantum Fisher information that can be extracted from it
by an optimally controlled probe state. In addition, we design an algorithm
that evaluates this quantum Fisher information via semidefinite programming. We
apply our framework to noisy frequency estimation, where we find that the
optimal performance of quantum metrology is better in the non-Markovian
scenario than in the Markovian scenario and explore the possibility of
efficient sensing via simple variational circuits.

Quantum Physics