Quantum trajectory approach to error mitigation

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 entails implementing the inverse of a known noise map of an environment by using a set of completely positive maps weighted by a quasiprobability distribution, i.e., a probability distribution with positive and negative values. This distribution is realized using classical postprocessing after final measurements of desired observables have been made. Here we make a connection with quasiprobability EM and recent results from quantum trajectory theory for open quantum systems. We show that the inverse of noise maps can be realized by performing classical postprocessing with a quasiprobability measure called the influence martingale on the quantum trajectories generated by an additional engineered reservoir coupled to the system. We demonstrate our result on a model relevant for current NISQ devices. Finally, we show the required quantum trajectories themselves can be simulated by coupling an ancillary qubit to the system. In this way, we can avoid the introduction of the engineered reservoir.