  Probabilistic Error Cancellation (PEC) aims to improve the accuracy of
expectation values for observables.This is accomplished using the probabilistic
insertion of recovery gates, which correspond to the inverse of errors.However,
the inserted recovery gates also induce errors. Thus, it is difficult to obtain
accurate expectation values with PEC since the estimator of PEC has a bias due
to noise induced by recovery gates.To address this challenge, we propose an
improved version of PEC that considers the noise resulting from gate insertion,
called Feed-Forward PEC (FFPEC). FFPEC provides an unbiased estimator of
expectation values by cancelling out the noise induced by recovery gates.We
demonstrate that FFPEC yields more accurate expectation values compared to
conventional PEC method through numerical simulations with bit-flip and
depolarizing noises.
