Recursive encoding and decoding of the noiseless subsystem for qudits

We give a full explanation of the noiseless subsystem that protects a single qubit against collective errors and the corresponding recursive scheme described by C.-K. Li et al. [Phys. Rev. A 84, 044301 (2011)] from a representation theory point of view. Furthermore, we extend the construction to qudits under the influence of collective SU($d$) errors. We find that under this recursive scheme, the asymptotic encoding rate is $1/d$.

Figure 1

Reordered version of the three-qubit QECC from [16]. A circuit representation for $U_E$ is shown in Fig. 2. $\left|\psi \right.〉$ represents the data qubit. $\left|v\right.〉$ is the state of the ancillary qubit, which can be arbitrary.

Figure 2

Reordered SU(2) encoding gate $U_E$ from [16], in terms of single-qubit and two-qubit controlled-$U$ gates. $Y_{\theta }=exp\left(i{\sigma }_y\theta \right)$ and $sin\Theta =\sqrt{2/3}$.

Figure 3

Recursive five-qubit circuit diagram from [16]. In this reordered version, the gates act on neighboring three-qubits only.

Figure 4

Schematic recursion. This reduced circuit is equivalent to the one shown in Fig. 3 due to the equivalence given in the text.

Figure 5

Final version of the circuit given in Fig. 3.