Perturbative quantum error correction

TL;DR

This paper establishes conditions for reversing quantum channels affected by small perturbations, using adapted Knill-Laflamme criteria, relevant for quantum error correction in weakly interacting systems.

Contribution

It extends quantum error correction theory by deriving conditions for reversing perturbed channels based on Lindblad operators and initial environmental states.

Findings

Reversal conditions are linked to Knill-Laflamme conditions applied to specific operator subspaces.

For weak environment interactions, the error space is determined by the initial environmental state.

The approach simplifies understanding of quantum error correction under small perturbations.

Abstract

We derive simple necessary and sufficient conditions under which a quantum channel obtained from an arbitrary perturbation from the identity can be reversed on a given code to the lowest order in fidelity. We find the usual Knill-Laflamme conditions applied to a certain operator subspace which, for a generic perturbation, is generated by the Lindblad operators. For a weak interaction with an environment, the error space to be corrected is a subspace of that spanned by the interaction operators, selected by the environment's initial state.