  Quantum Error Mitigation (EM) is a collection of strategies to reduce errors
on noisy intermediate scale quantum (NISQ) devices on which proper quantum
error correction is not feasible. One of such strategies aimed at mitigating
noise effects of a known environment is to realise the inverse map of the noise
using a set of completely positive maps weighted by a quasi-probability
distribution, i.e. a probability distribution with positive and negative
values. This quasi-probability distribution is realised using classical
post-processing after final measurements of the desired observables have been
made. Here we make a connection with quasi-probability EM and recent results
from quantum trajectory theory for open quantum systems. We show that the
inverse of noise maps can be realised by performing classical post-processing
on the quantum trajectories generated by an additional reservoir with a
quasi-probability measure called the influence martingale. We demonstrate our
result on a model relevant for current NISQ devices. Finally, we show the
quantum trajectories required for error correction can themselves be simulated
by coupling an ancillary qubit to the system. In this way, we can avoid the
introduction of the engineered reservoir.

Quantum Physics