Editing Quantum error correction (section)

== Bosonic codes ==
Several proposals have been made for storing error-correctable quantum information in bosonic modes.{{Clarification needed|date=May 2022|reason=Clarify why they are called bosonic codes, what they have to do with bosons.}} Unlike a two-level system, a [[quantum harmonic oscillator]] has infinitely many energy levels in a single physical system. Codes for these systems include cat,<ref name=":2">{{Cite journal| last1=Cochrane| first1=P. T. |last2=Milburn| first2=G. J.| last3=Munro| first3=W. J.| date=1999-04-01| title=Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping| journal=Physical Review A| volume=59| issue=4| pages=2631–2634| doi=10.1103/PhysRevA.59.2631| arxiv=quant-ph/9809037| bibcode=1999PhRvA..59.2631C| s2cid=119532538}}</ref><ref name=":3">{{Cite journal| last1=Leghtas| first1=Zaki| last2=Kirchmair| first2=Gerhard| last3=Vlastakis| first3=Brian| last4=Schoelkopf| first4=Robert J.| last5=Devoret| first5=Michel H.| last6=Mirrahimi| first6=Mazyar| date=2013-09-20| title=Hardware-Efficient Autonomous Quantum Memory Protection| journal=Physical Review Letters| volume=111| issue=12| pages=120501| doi=10.1103/physrevlett.111.120501| pmid=24093235| arxiv=1207.0679| s2cid=19929020| bibcode=2013PhRvL.111l0501L| issn=0031-9007}}</ref><ref name=":4">{{Cite journal| last1=Mirrahimi| first1=Mazyar| last2=Leghtas| first2=Zaki| last3=Albert| first3=Victor V| last4=Touzard| first4=Steven| last5=Schoelkopf| first5=Robert J| last6=Jiang| first6=Liang| last7=Devoret| first7=Michel H| date=2014-04-22| title=Dynamically protected cat-qubits: a new paradigm for universal quantum computation|journal=New Journal of Physics| volume=16| issue=4| pages=045014| doi=10.1088/1367-2630/16/4/045014| arxiv=1312.2017| bibcode=2014NJPh...16d5014M| s2cid=7179816| issn=1367-2630}}</ref> Gottesman-Kitaev-Preskill (GKP),<ref>{{Cite journal| arxiv=quant-ph/0008040| author1=Daniel Gottesman| author2=Alexei Kitaev| author3=John Preskill| title=Encoding a qubit in an oscillator| journal=Physical Review A| volume=64| issue=1| pages=012310| doi=10.1103/PhysRevA.64.012310| bibcode=2001PhRvA..64a2310G| year=2001| s2cid=18995200}}</ref> and binomial codes.<ref name=":0">{{Cite journal| last1=Michael| first1=Marios H.| last2=Silveri| first2=Matti| last3=Brierley| first3=R. T.| last4=Albert| first4=Victor V.| last5=Salmilehto| first5=Juha| last6=Jiang| first6=Liang| last7=Girvin| first7=S. M.| date=2016-07-14| title=New Class of Quantum Error-Correcting Codes for a Bosonic Mode| journal=Physical Review X| volume=6| issue=3| pages=031006| doi=10.1103/PhysRevX.6.031006 |arxiv=1602.00008| bibcode=2016PhRvX...6c1006M| s2cid=29518512}}</ref><ref>{{Cite journal| first1=Victor V. |last1=Albert| first2=Kyungjoo |last2=Noh| first3=Kasper |last3=Duivenvoorden| first4=Dylan J. |last4=Young| first5=R. T. |last5=Brierley| first6=Philip |last6=Reinhold| first7=Christophe |last7=Vuillot| first8=Linshu |last8=Li| first9=Chao |last9=Shen| first10=S. M. |last10=Girvin| first11=Barbara M. |last11=Terhal| first12=Liang |last12=Jiang| year=2018| title=Performance and structure of single-mode bosonic codes| journal=Physical Review A| volume=97| issue=3| pages=032346| arxiv=1708.05010| s2cid=51691343| bibcode=2018PhRvA..97c2346A| doi=10.1103/PhysRevA.97.032346}}</ref> One insight offered by these codes is to take advantage of the redundancy within a single system, rather than to duplicate many two-level qubits.

=== 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.

===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" />