Quantum error correction with mixed ancilla qubits

Most quantum error-correcting codes are predicated on the assumption that there exists a reservoir of qubits in the state $|0\rangle$ which can be used as ancilla qubits to prepare multiqubit logical states. In this Brief Report, we examine the consequences of relaxing this assumption and propose a method to increase the fidelity produced by a given code when the ancilla qubits are initialized in mixed states, using the same number of qubits, at most doubling the number of gates when the recovery operation would already be implemented. The procedure implemented consists of altering the encoding operator to include the inverse of the unitary operation used to correct detected errors after decoding. This augmentation will be especially useful in quantum computing architectures that do not possess projective measurement, such as solid-state NMR quantum information processing.

Figure 1

(top) The traditional three-qubit code to correct bit-flip errors and (bottom) that augmented to provide increased fidelity in the case where each ancilla qubit is subject to the initialization noise map discussed above. The map $E$ in this example is the bit-flip map $\{\sqrt{1-p}\widehat{I},\sqrt{p}\widehat{X}\}$. The augmentation consists of implementing the Toffoli gate used to correct detected errors before the standard encoding procedure takes place. This improves the overall fidelity by ensuring that, if no error occurs, the encoded state remains unaltered by false syndromes.

Figure 2

In order to augment an error-correction code on $n$ qubits, the recovery operator is inverted and implemented before encoding. This eliminates faults caused solely by false syndromes. Here, $R$ is the recovery operator, $C$ is the encoding operator, and $E$ is the error map.

Figure 3

The initialization error for which an error-correcting code can give a channel fidelity $\ge$$1-p$. This is shown for four repetition codes correcting bit-flip errors. Note that the tolerable error for small values of $p$, the parameter describing the main bit-flip channel, approaches 0 rapidly for unaugmented codes. In contrast, augmentation provides a high tolerable $q$ for every value of $p$.

Figure 4

The initialization error for which an error-correcting code can give a channel fidelity $\ge 1-\frac{3}{4}p$. This is shown for the perfect five-qubit code. Note that the behavior of this code is qualitatively different, having zero tolerable initialization for $p\sim 0.18$. The ability of the augmented code to provide finite tolerable $q$ at $p=0$ is preserved.

Figure 5

The tolerable initialization noise for different concatenated codes. For the top-level concatenation, the encoder used in the top panel of Fig. 1 has been replaced with the encoder used in the bottom panel of Fig. 1. For the full concatenation, all the encoding circuits used are augmented, as in Fig. 1. Note that the bottom curve (for the unaugmented concatenated code) is identical to the tolerable initialization noise for the three-bit error correcting code when left unconcatenated, shown in Fig. 3. Also, the tolerable $q$ for the fully augmented code is $2-\sqrt{2}$, identical to the augmented three-qubit code.

Figure 6

The augmented version of the “perfect” five-qubit code given in [14] and the errata to [14]. The top circuit is the unaugmented encoder, and the bottom circuit is the correction operator. These are combined according to the prescription in Fig. 2. Here, the error channel is the depolarizing channel $\{\sqrt{1-3p/4}\widehat{I},\sqrt{p/4}\widehat{X},\sqrt{p/4}\widehat{Y},\sqrt{p/4}\widehat{Z}\}$. The augmentation has a similar effect to that used on the $(2t+1)$-qubit codes countering $t\text{th}$-order bit-flip errors. We can deduce from this that the benefits of augmentation as described above are not limited to codes which counter classical errors.