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符号翻转码

编码

原始形式:

0+++1\begin{array}{l}|0\rangle \rightarrow|+++\rangle \\ |1\rangle \rightarrow|---\rangle\end{array}

经过处理的形式:

012(000+011+101+111112(111+100+010+001\begin{aligned}|0\rangle & \rightarrow \frac{1}{2}(|000\rangle+|011\rangle+|101\rangle+|111\rangle \\ |1\rangle & \rightarrow \frac{1}{2}(|111\rangle+|100\rangle+|010\rangle+|001\rangle\end{aligned}

其规律为:

  • 将 0 编码为偶数个 1,1 编码为奇数个 1

解码

投影到以下四个子空间:

0L0L+1L1Lσz(j)0L0Lσzj+σzj1L1Lσzj\begin{array}{c}\left|0_{L}\right\rangle\left\langle 0_{L}|+| 1_{L}\right\rangle\left\langle 1_{L}\right| \\ \sigma_{z}^{(j)}\left|0_{L}\right\rangle\left\langle 0_{L}\left|\sigma_{z}^{j}+\sigma_{z}^{j}\right| 1_{L}\right\rangle\left\langle 1_{L}\right| \sigma_{z}^{j}\end{array}