Editing Quantum error correction (section)

===Cat code===
[[Cat state|Schrödinger cat states]], superpositions of coherent states, can also be used as logical states for error correction codes. Cat code, realized by Ofek et al.<ref name=":5" /> in 2016, defined two sets of logical states: <math>\{|0^+_L\rangle, |1^+_L\rangle\} </math> and <math>\{|0^-_L\rangle, |1^-_L\rangle\} </math>, where each of the states is a superposition of [[coherent state]] as follows

<math display="block">\begin{aligned}
    |0^+_L\rangle& \equiv |\alpha\rangle + |-\alpha\rangle, \\
    |1^+_L\rangle& \equiv |i\alpha\rangle + |-i\alpha\rangle, \\
    |0^-_L\rangle& \equiv |\alpha\rangle - |-\alpha\rangle, \\
    |1^-_L\rangle& \equiv |i\alpha\rangle - |-i\alpha\rangle.
\end{aligned}</math>

Those two sets of states differ from the photon number parity, as states denoted with <math>^+</math> only occupy even photon number states and states with <math>^-</math> indicate they have odd parity. Similar to the binomial code, if the dominant error mechanism of the system is the stochastic application of the bosonic [[lowering operator]] <math>\hat{a}</math>, the error takes the logical states from the even parity subspace to the odd one, and vice versa. Single-photon-loss errors can therefore be detected by measuring the photon number parity operator <math>\exp(i\pi \hat{a}^\dagger\hat{a}) </math> using a dispersively coupled ancillary qubit.<ref name=nature13436/>

Still, cat qubits are not protected against two-photon loss <math>\hat{a}^2</math>,  dephasing noise <math>\hat{a}^\dagger\hat{a}</math>, photon-gain error <math>\hat{a}^\dagger</math>, etc.<ref name=":2" /><ref name=":3" /><ref name=":4" />