  Distinct non-Hermitian dynamics has demonstrated its advantages in improving
measurement precision over traditional sensing protocols. Multi-mode
non-Hermitian lattice dynamics can provide exponentially-enhanced quantum
sensing where the quantum Fisher information (QFI) per photon increases
exponentially with the lattice size. However, somewhat surprisingly, it was
also shown that the quintessential non-Hermitian skin effect does not provide
any true advantage. In this paper, we demonstrate the importance of optimizing
the phase of the coherent drive, and the position of the injection and
detection in multi-mode non-Hermitian quantum sensing. The QFI per photon can
be exponentially-enhanced or exponentially-reduced depending on parameters of
the drive and detection. Specifically, it is demonstrated that for large
amplification by choosing appropriate coherent drive parameters, the
non-Hermitian skin effect can provide exponentially-enhanced quantum sensing.
Moreover, in the regime beyond linear response, skin-effect can also provide a
dramatic advantage as compared to the local perturbation, and the proposed
protocol is robust in tuning the amplification factor.
