On the probabilistic quantum error correction

TL;DR

This paper analyzes probabilistic quantum error correction, extending conditions for success, exploring encoding strategies, and demonstrating advantages over deterministic methods for broader noise correction.

Contribution

It generalizes Knill-Laflamme conditions for probabilistic error correction and shows how to optimize encoding to improve success probability.

Findings

Encoding into mixed states can maximize correction success.

Probabilistic correction can handle broader noise classes.

Single auxiliary qubit suffices for unitary errors.

Abstract

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 deeply analyze probabilistic version of the error-correcting procedure for general noise. We generalized the Knill-Laflamme conditions for probabilistically correctable errors. We show that for some noise channels, we should encode the information into a mixed state to maximize the probability of successful error correction. Finally, we investigate an advantage of the probabilistic error-correcting procedure over the deterministic one. 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.