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-component 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 \tag*{(1)}\end{align} is the corresponding pair-coherent state [24] 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-component 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 \(\gamma^2\).


Microwave cavities coupled to superconducting circuits by the Wang group [6].



  • 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.
  • Quantum spherical code (QSC) — Pair-cat codes are QSCs embedded into the configuration space of pair-coherent states.


V. V. Albert et al., “Pair-cat codes: autonomous error-correction with low-order nonlinearity”, Quantum Science and Technology 4, 035007 (2019) arXiv:1801.05897 DOI
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
M. Yuan, Q. Xu, and L. Jiang, “Construction of bias-preserving operations for pair-cat codes”, Physical Review A 106, (2022) arXiv:2208.06913 DOI
J. M. Gertler et al., “Experimental Realization and Characterization of Stabilized Pair Coherent States”, (2022) arXiv:2209.11643
Page edit log

Your contribution is welcome!

on (edit & pull request)— see instructions

edit on this site

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={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

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