# Pair-cat code[1]

## Description

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.

## Protection

## Gates

## Decoding

## Parent

## Cousins

- 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.

## References

- [1]
- 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
- [2]
- A. O. Barut and L. Girardello, “New “Coherent” States associated with non-compact groups”, Communications in Mathematical Physics 21, 41 (1971). DOI
- [3]
- 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
- [4]
- G. S. Agarwal, “Nonclassical statistics of fields in pair coherent states”, Journal of the Optical Society of America B 5, 1940 (1988). DOI
- [5]
- Ming Yuan, Qian Xu, and Liang Jiang, “Construction of Bias-preserving Operations for Pair-cat Code”. 2208.06913

## Zoo code information

## Cite as:

“Pair-cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/paircat