On the Probabilistic Quantum Error Correction

Probabilistic quantum error correction is an error-correcting procedure which uses postselection to determine if the encoded information was successfully restored. In this work, we analyze the probabilistic version of the error-correcting procedure for general noise. We generalize the Knill-Laflamme conditions for probabilistically correctable errors. We show that for some noise channels, the initial information has to be encoded into a mixed state to maximize the probability of successful error correction. Finally, the probabilistic error-correcting procedure offers an advantage over the deterministic procedure. Reducing the probability of successful error correction allows for correcting errors generated by a broader class of noise channels. Significantly, if the errors are caused by a unitary interaction with an auxiliary qubit system, we can probabilistically restore a qubit state by using only one additional physical qubit.