# On the Performance of Interleavers for Quantum Turbo Codes
## 2 Classical Interleavers for Quantum Turbo Codes  
### System Model  
The Quantum Turbo Codes considered in this paper consist of the interleaved serial concatenation of unassisted QCCs acting as outer codes and entanglement-assisted QCCs inner codes. Figure 1 presents the full schematic representation of such quantum error correction system. The \(k\) input logical qubits that compose the information word \(|\psi_{1}\rangle\) are first fed to the outer \([n_{1},k_{1},m_{1}]\) unassisted convolutional encoder \(\mathcal{V}_{1}\) and encoded into \(n^{\prime}=k\frac{n_{1}}{k_{1}}\) physical qubits with the help of \((n_{1}-k_{1})\) ancilla qubits and \(m\) memory qubits. The \(n^{\prime}\) physical qubits that form the codeword \(|\hat{p}_{1}\rangle\) generated by the first encoder are then passed through a quantum interleaver \(\Pi\) before being input to the inner convolutional encoder \(\mathcal{V}_{2}\). Such an encoder is an \([n_{2},k_{2},m_{2},c]\) entanglement-assisted encoder that encodes the interleaved sequence of \(n^{\prime}\) qubits \(|\psi_{2}\rangle\) into the codeword \(|\hat{p}_{2}\rangle\) of length \(n=n^{\prime}\frac{n_{2}}{k_{2}}=k\frac{n_{2}}{k_{2}}\frac{n_{1}}{k_{1}}\), aided by \((n_{2}-k_{2}-c)\) ancilla qubits, \(c\) pre-shared EPR pairs and \(m_{2}\) memory qubits. Codeword \(|\hat{p}_{2}\rangle\) is then transmitted through a quantum depolarizing channel (The depolarizing channel is a widely used channel model in order to represent the decoherence effects that produce errors in quantum information [3; 4; 5; 6; 7].) with depolarizing probability \(p\) inflicting an \(n\)-qubit Pauli error \(\mathcal{P}_{2}\in\mathcal{G}_{n}\) to the codeword. The operation of the depolarizing channel on an individual qubit with density matrix \(\rho\) is defined as  
\[\mathcal{N}_{D}(\rho)=(1-p)\rho+\frac{p}{3}(\mathbf{X}\rho\mathbf{X}+\mathbf{ Y}\rho\mathbf{Y}+\mathbf{Z}\rho\mathbf{Z}), \tag{1}\]  
where \(p\) is the depolarizing probability and \(\mathbf{X},\mathbf{Y},\mathbf{Z}\) are the Pauli matrices. The depolarizing channel is independently applied to each of the qubits of the stream \(|\hat{p}_{2}\rangle\), and, consequently, each of the qubits experiences a bit-flip (\(\mathbf{X}\) operator) with probability \(p/3\), a phase-flip (\(\mathbf{Z}\) operator) with probability \(p/3\) or a combination of both (\(\mathbf{Y}\) operator) with probability \(p/3\). At the output of the depolarizing channel, the state \(\mathcal{P}_{2}\ket{\bar{\psi}_{2}}\) is fed to the inverse of the inner encoder \(\mathcal{V}_{2}^{\dagger}\), which outputs the decoded state \(\mathcal{L}_{2}\ket{\psi_{2}}\), where \(\mathcal{L}_{2}\in\mathcal{G}_{n^{\prime}}\) refers to the logical error suffered by the decoded state due to the operation of the channel; and the classical syndrome bits \(\mathrm{R}_{2}=(\mathrm{S}_{2}^{x},\mathrm{E}_{2}^{x,z})\) obtained from \(Z\) basis measurements on the ancilla qubits and Bell measurements on the pre-shared EPR pairs. The corrupted logical qubits are then passed through a de-interleaver \(\Pi^{-1}\) resulting in the state \(\mathcal{P}_{1}\ket{\bar{\psi}_{1}}\), which is supplied to the inverse of the outer encoder \(\mathcal{V}_{1}^{\dagger}\). The resulting output is the state \(\mathcal{L}_{1}\ket{\psi_{1}}\), which corresponds to the information quantum state corrupted by a logical error \(\mathcal{L}_{1}\in\mathcal{G}_{k}\); and the classical syndrome bits \(\mathrm{S}_{1}^{x}\) obtained after measuring the ancilla qubits on the \(Z\) basis.