Numerically optimized bosonic code[1,2] 

Bosonic Fock-state code obtained from a numerical minimization procedure, e.g., from enforcing error-correction criteria against some number of losses while minimizing average occupation number. Useful single-mode codes can be determined using basic numerical optimization [1,2], semidefinite-program recovery/encoding optimization [3,4], or reinforcement learning [5,6].

The smallest numerically optimized code protecting against a single loss error is the \(\sqrt(17)\) code [1], \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{6}}\left(\sqrt{7-\sqrt{17}}|0\rangle+\sqrt{\sqrt{17}-1}|3\rangle\right)\\ |\overline{1}\rangle&=\frac{1}{\sqrt{6}}\left(\sqrt{9-\sqrt{17}}|1\rangle-\sqrt{\sqrt{17}-3}|4\rangle\right)~, \end{split} \tag*{(1)}\end{align} correcting a single loss error. The average occupation number of the codewords is \(\approx 1.6\), which is \(0.4\) photons lower than that of the smallest binomial code with the same level of protection.