Quantum error correction against correlated noise

We consider quantum error correction against correlated noise using simple and concatenated Calderbank-Shor-Steane codes as well as $n$-qubit repetition codes. We characterize the performance of various codes by means of the fidelity following the error correction of a single logical qubit in a quantum register. For concatenated codes we find a threshold in the single-qubit error rate below which the encoded qubit is perfectly protected. The threshold depends on the correlation strength of the noise and goes to zero for perfect correlation. Finally, we concatenate the traditional error correcting codes with a decoherence free subspace and evaluate the performance over the whole range from uncorrelated noise to perfectly correlated noise.

Figure 1

The enhancement of the failure rate for the three-qubit repetition code (i), the $⟦7,1,3⟧$ code (ii), and the $⟦23,1,7⟧$ code (iii). The full curve for the $⟦23,1,7⟧$ code is plotted in the inset.

Figure 2

The fidelity of the three-qubit repetition code (a), the $⟦7,1,3⟧$ code (b), and the $⟦23,1,7⟧$ code (c).

Figure 3

The threshold curve for the three qubit code (a) for $j=1$ (i), 2 (ii), and 3 (iii). The threshold curves (b) for the $⟦7,1,3⟧$ code for $j=1$ (i) and 2 (ii) and for the $⟦23,1,7⟧$ code for $j=1$ (iii).

Figure 4

A set of curves where the fidelity of the $n$-bit repetition code is equal to the fidelity of a single qubit. Shown for $n=3,5,\dots ,15$, with $n=3$ the uppermost curve and $n=15$ the lowermost curve.

Figure 5

The fidelity of the bare DFS (a) and the three-bit code concatenated with the DFS (b).

Figure 6

The threshold curve for the three-bit code concatenated with DFS’s (solid line). The threshold for the three-bit code without DFS’s (dashed line).