Post-selected PI code[1]

Description

PI qubit code whose recovery succeeds at protecting against AD errors with a success probability less than one.

The simplest code admits a codeword basis of \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{3}}\left(|100\rangle+|010\rangle+|001\rangle\right)\\|\overline{1}\rangle&=|111\rangle~. \end{split} \tag*{(1)}\end{align} The code violates the diagonal part of the Knill-Laflamme conditions. Nevertheless, the code admits a probabilistic recovery that protects against single losses and yields an infidelity of order \(O(\gamma^2)\) in the noise rate \(\gamma\). The failure probability of the recovery is of the same order as the probability of the single loss errors, i.e., \(O(\gamma)\).