Pair-cat code[1]


Two- or higher-mode extension of cat codes whose codewords are right eigenstates of powers of products of the modes' lowering operators. Many gadgets for cat codes have two-mode pair-cat analogues, with the advantage being that such gates can be done in parallel with a dissipative error-correction process.

Two-mode codewords are supported by Fock states with occupation number \(\hat{n}_2-\hat{n}_1\) fixed to some integer \(\Delta\). In the two-legged case, \(|\overline{0}_{\gamma,\Delta}\rangle \sim |\gamma_\Delta \rangle + (-1)^\Delta |i\gamma_\Delta\rangle\) and \(|\overline{1}_{\gamma,\Delta}\rangle \sim |\gamma_\Delta\rangle - (-1)^\Delta |i \gamma\rangle\), where \begin{align} |\alpha_\Delta \rangle \propto \sum_{n=0}^\infty \frac{\alpha^{2n+\Delta}}{\sqrt{n! (n+\Delta)!}} |n,n+\Delta\rangle \end{align} is the corresponding pair-coherent state [2][3][4] with complex amplitude \(\alpha\), up to normalization.


The occupation-number differences form the syndromes, as opposed to the photon number parity for the single-mode cat code. Any loss even combination that changes the relative differences of photons between modes is a detectable error. The two-mode two-legged paircat code can detect arbitrary single-mode losses, but cannot detect simultaneous photon loss in both modes. An \(n\)-mode code can detect any loss errors of at most \(n-1\) weight. Higher numbers of legs correspond to more pair-coherent state present in the codewords, and allow for protection against simulataneous losses.


Hamiltonian \(X\), \(XX\), \(Z\) gates, holonomic \(Z\) gate, control-phase gate.Bias-preserving gates [5].


Lindbladian-based dissipative encoding utilizing two-mode two-photon absorption [3]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(\gamma^2\).



  • Cat code — Cat (pair-cat) codewords are superpositions of coherent (pair-coherent) states. Many cat-code protocols have analogues for the two-mode pair-cat codes.
  • Hamiltonian-based code — Two-legged pair-cat codewords form ground-state subspace of a multimode Kerr Hamiltonian.


V. V. Albert et al., “Pair-cat codes: autonomous error-correction with low-order nonlinearity”, Quantum Science and Technology 4, 035007 (2019). DOI; 1801.05897
A. O. Barut and L. Girardello, “New “Coherent” States associated with non-compact groups”, Communications in Mathematical Physics 21, 41 (1971). DOI
G. S. Agarwal, “Generation of Pair Coherent States and Squeezing via the Competition of Four-Wave Mixing and Amplified Spontaneous Emission”, Physical Review Letters 57, 827 (1986). DOI
G. S. Agarwal, “Nonclassical statistics of fields in pair coherent states”, Journal of the Optical Society of America B 5, 1940 (1988). DOI
Ming Yuan, Qian Xu, and Liang Jiang, “Construction of Bias-preserving Operations for Pair-cat Code”. 2208.06913

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Internal code ID: paircat

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Zoo Code ID: paircat

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

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