Editing Quantum error correction (section)

=== Binomial code ===
Written in the [[Fock state|Fock]] basis, the simplest binomial encoding is
<math display="block">|0_{\rm L}\rangle=\frac{|0\rangle+|4\rangle}{\sqrt{2}},\quad |1_{\rm L}\rangle=|2\rangle,</math>
where the subscript L indicates a "logically encoded" state. Then if the dominant error mechanism of the system is the stochastic application of the bosonic [[lowering operator]] <math>\hat{a},</math> the corresponding error states are <math>|3\rangle</math> and <math>|1\rangle,</math> respectively. Since the codewords involve only even photon number, and the error states involve only odd photon number, errors can be detected by measuring the [[photon number]] parity of the system.<ref name=":0" /><ref name=nature13436>{{Cite journal| last1=Sun| first1=L.| last2=Petrenko| first2=A.| last3=Leghtas| first3=Z.| last4=Vlastakis| first4=B.| last5=Kirchmair| first5=G.| last6=Sliwa| first6=K. M.| last7=Narla| first7=A.| last8=Hatridge| first8=M.| last9=Shankar| first9=S.| last10=Blumoff| first10=J.| last11=Frunzio| first11=L.| last12=Mirrahimi| first12=M.| last13=Devoret| first13=M. H.| last14=Schoelkopf| first14=R. J.| date=July 2014| title=Tracking photon jumps with repeated quantum non-demolition parity measurements| journal=Nature| language=en| volume=511| issue=7510| pages=444–448| doi=10.1038/nature13436| pmid=25043007| issn=1476-4687| arxiv=1311.2534| bibcode=2014Natur.511..444S| s2cid=987945}}</ref> Measuring the odd parity will allow correction by application of an appropriate unitary operation without knowledge of the specific logical state of the qubit. However, the particular binomial code above is not robust to two-photon loss.