Nonlinear quantum error correction

We introduce a theory of quantum error correction (QEC) for a subclass of states. 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.

Figure 1

State structure for nonlinear quantum error correction. The set of alphabet states $\left|\psi \right(\mathit{\alpha })\rangle$ is depicted as the darkened area embedded in a subspace defined by the projector $P$. For ${\mathrm{\Gamma }}_{nm}=0$, the two errors $F_n$ and $F_m$ map the states to orthogonal subspaces. In the figure we have ${\mathrm{\Gamma }}_{01}={\mathrm{\Gamma }}_{02}=0$. For ${\mathrm{\Gamma }}_{nm}=1$, the two errors map states to the same subspace, we thus have ${\mathrm{\Gamma }}_{12}=1$ for the example above. The alphabet states with errors $F_n|\psi \left(\mathit{\alpha }\right)\rangle$ again constitute a smaller set of these subspaces, represented by the darkened areas, embedded in larger subspaces defined by $P_n$. Correction operations $U_n^{†}$ return the state to its original state, completing the error correction. In practice, an error of type $E_n|\psi \left(\mathit{\alpha }\right)\rangle$ can also be handled, which corresponds to a superposition across different spaces $P_n$. Projection of $E_n|\psi \left(\mathit{\alpha }\right)\rangle$ by $P_m$ collapses the state to $F_m|\psi \left(\mathit{\alpha }\right)\rangle$, after which the correction operation is identical.