Nonlinear quantum error correction

TL;DR

This paper develops a new framework for quantum error correction that applies to specific subclasses of states, relaxing traditional linear constraints and enabling error correction in scenarios previously deemed impossible, such as Gaussian states.

Contribution

It introduces nonlinear quantum error correction (NLQEC), providing a sufficiency criterion for correcting errors within subclasses of states, expanding the scope of QEC beyond linear code spaces.

Findings

NLQEC offers a more relaxed error correction condition.

It circumvents no-go theorems for optical QEC with Gaussian states.

Explicit examples demonstrate the effectiveness of NLQEC.

Abstract

We introduce a theory of quantum error correction (QEC) for a subclass of states within a larger Hilbert space. In the standard theory of QEC, the set of all encoded states is formed by an arbitrary linear combination of the codewords. However, this can be more general than required for a given quantum protocol which may only traverse a subclass of states within the Hilbert space. Here we propose the concept of nonlinear QEC (NLQEC), where the encoded states are not necessarily a linear combination of codewords. We introduce a sufficiency criterion for NLQEC with respect to the subclass of states. The new criterion gives a more relaxed condition for the formation of a QEC code, such that under the assumption that the states are within the subclass of states, the errors are correctable. This allows us, for instance, to effectively circumvent the no-go theorems regarding optical QEC for Gaussian states and channels, for which we present explicit examples.