We study efficient quantum error correction schemes for the fully correlated channel on an n-qubit system with error operators that assume the form \(\sigma _x^{\otimes n}\), \(\sigma _y^{\otimes n}\), \(\sigma _z^{\otimes n}\). Previous schemes are improved to facilitate implementation. In particular, when n is odd and equals \(2k+1\), we describe a quantum error correction scheme using one arbitrary qubit \(\sigma \) to protect the data state \(\rho \) in a 2k-qubit system. The encoding operation \(\sigma \otimes \rho \mapsto \Phi (\sigma \otimes \rho )\) only requires 3k CNOT gates (each with one control bit and one target bit). After the encoded state \(\Phi (\sigma \otimes \rho )\) goes through the channel, we can apply the inverse operation \(\Phi ^{-1}\) to produce \({\tilde{\sigma }} \otimes \rho \) so that a partial trace operation can recover \(\rho \). When n is even and equals \(2k+2\), we describe a hybrid quantum error correction scheme using any one of the two classical bits \(\sigma \in \{|ij{\rangle }{\langle }ij|: i, j \in \{0,1\}\}\) to protect a 2k-qubit state \(\rho \) and two classical bits. The encoding operation \(\sigma \otimes \rho \mapsto \Phi (\sigma \otimes \rho )\) can be done by \(3k+2\) CNOT gates and a single-qubit Hadamard gate. After the encoded state \(\Phi (\sigma \otimes \rho )\) goes through the channel, we can apply the inverse operation \(\Phi ^{-1}\) to produce \(\sigma \otimes \rho \) so that a perfect protection of the two classical bits \(\sigma \) and the 2k-qubit state is achieved. If one uses an arbitrary two-qubit state \(\sigma \), the same scheme will protect 2k-qubit states. The scheme was implemented using MATLAB, Mathematica, Python and the IBM’s quantum computing framework qiskit.

中文翻译：

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完全相关的量子通道的纠错方案，可同时保护量子信息和经典信息

我们研究具有误差算子的n -qubit系统上全相关信道的有效量子纠错方案，这些算子的形式为\（\ sigma _x ^ {\ otimes n} \），\（\ sigma _y ^ {\ otimes n} \），\（\ sigma _z ^ {\ otimes n} \）。改进了先前的方案以促进实施。特别是，当Ñ是奇数且等于\第（2k + 1 \） ，我们使用一个任意的量子位描述一个量子纠错方案\（\西格玛\）以保护数据状态\（\ RHO \）在一个2 ķ -量子位系统。编码操作\（\ sigma \ otimes \ rho \ mapsto \ Phi（\ sigma \ otimes \ rho）\）仅需要3k个CNOT门（每个都有1个控制位和1个目标位）。编码状态\（\ Phi（\ sigma \ otimes \ rho）\）通过通道后，我们可以应用逆运算\（\ Phi ^ {-1} \）来生成\（{\ tilde {\ sigma }} \ otimes \ rho \）以便部分跟踪操作可以恢复\（\ rho \）。当n为偶数且等于\（2k + 2 \）时，我们使用两个经典位\（\ sigma \ in \ {| ij {\ rangle} {\ langle ijij |中的任意一个来描述一种混合量子纠错方案。 ：i，j \ in \ {0,1 \} \} \）保护2 k -qubit状态\（\ rho \）和两个经典位。编码操作\（\ sigma \ otimes \ rho \ mapsto \ Phi（\ sigma \ otimes \ rho）\）可以通过\（3k + 2 \） CNOT门和单量子位Hadamard门来完成。编码状态\（\ Phi（\ sigma \ otimes \ rho）\）通过通道后，我们可以应用逆运算\（\ Phi ^ {-1} \）来生成\（\ sigma \ otimes \ rho \），因此可以完美保护两个经典位\（\ sigma \）和2 k -qubit状态。如果使用任意两个量子位状态\（\ sigma \），则相同的方案将保护2 k个量子位状态。该方案是使用MATLAB，Mathematica，Python和IBM量子计算框架qiskit实施的。