  Noise is the main source that hinders us from fully exploiting quantum
advantages in various quantum informational tasks. However, characterizing and
calibrating the effect of noise is not always feasible in practice. Especially
for quantum parameter estimation, an estimator constructed without precise
knowledge of noise entails an inevitable bias. Recently, virtual
purification-based error mitigation (VPEM) has been proposed to apply for
quantum metrology to reduce such a bias occurring from unknown noise. While it
was demonstrated to work for particular cases, whether VPEM always reduces a
bias for general estimation schemes is unclear yet. For more general
applications of VPEM to quantum metrology, we study factors determining whether
VPEM can reduce the bias. We find that the closeness between the dominant
eigenvector of a noisy state and the ideal quantum probe (without noise) with
respect to an observable determines the reducible amount of bias by VPEM. Next,
we show that one should carefully choose the reference point of the target
parameter, which gives a smaller bias than others because the bias depends on
the reference point. Otherwise, even if the dominant eigenvector and the ideal
quantum probe are close, the bias of the mitigated case could be larger than
the non-mitigated one. Finally, we analyze the error mitigation for a phase
estimation scheme under various noises. Based on our analysis, we predict
whether VPEM can effectively reduce a bias and numerically verify our results.
