Cat code[1][2]


Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator. Codewords for a qubit code (\(q=2\)) consist of a coherent state \(|\alpha\rangle\) projected onto a subspace of Fock state number modulo \(2(S+1)\). The logical state \(|\overline{0}\rangle\) is in the \(\{|0\rangle , |2(S+1)\rangle , |4(S+1)\rangle \cdots \}\) Fock-state subspace, while \(|\overline{1}\rangle\) is in the \(\{|(S+1)\rangle, |3(S+1)\rangle , |5(S+1)\rangle , |7(S+1)\rangle \cdots \}\) subspace. An alternative basis, valid for for general \(q\) and \(\alpha\neq 0\), consists of \(q\) coherent states distributed equidistanctly around a circle in phase space of radius \(\alpha\).


Due to the spacing between sets of Fock states, the distance between two distinct logical states is \(d=S+1\). Hence, this code is able to detect \(S\)-photon loss error. Two-legged cat codes (\(S=1\)) do not protect against loss events, but there exist modifications based on sign alternation [3] or squeezing [4] that add such protection.


Lindbladian-based dissipative encoding utilizing multi-photon absorption [5].Hamiltonian-based 'Kerr-cat' encoding utilizing the Kerr effect [6] (see also Ref. [7]).Combined dissipative and Hamiltonian-based encoding utilizing two-photon exchange for \(S=1\) codes [8].


For \(S=1\), universal gates can be performed using displacement operators and a rotation based on the Kerr nonlinearity [5]. For \(S=2\), squeezing replaces displacements.Holonomic gates utilizing the Berry phase of coherent states are universal [9].Bias-preserving CNOT gate utilizing a topological Berry phase [10].


Measurement in the Fock basis. For a \(2(S+1)\) cat code, a number measurement returns outcome \(2(S+1)k\), if \(k\) is even, then it corresponds to logical 0 state; if \(k\) is odd, then it corresponds to logical 1 state.

Fault Tolerance

Bias-preserving CNOT gate [10] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [11][10].


Two-legged (\(S=1\)) cat code has been realized in a superconducting circuit device by the Devoret group [12]. Exponential suppression of bit-flip errors achieved [13] up to a bit-flip time of 1 ms. A bit-flip time of up to 1 sec has been achieved while away from the logical-qubit regime [14].Four-legged (\(S=2\)) cat code has been realized in a superconducting circuit device [15]. This paper is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the lifetime of the noisiest constituent of the system.


Pedagogical introduction to cat codes can be found in Ref. [16].


  • Bosonic rotation code — The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) [17].


  • Number-phase code — In the limit as \(N,S \to \infty\), phase measurement in the cat code has vanishing variance, just like in a number-phase code [17].
  • Hamiltonian-based code — Two-legged cat codewords form ground-state subspace of a Kerr Hamiltonian [6].
  • Binomial code — For a fixed \(S\), binomial codes with \(N \to \infty\) coincide with cat codes as \(\alpha \to \infty\) [18].

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Internal code ID: cat

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Cite as:
“Cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_cat, title={Cat code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

“Cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.