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

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.

Gates

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

Decoding

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

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

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. https://errorcorrectionzoo.org/c/paircat
BibTeX:
@incollection{eczoo_paircat, title={Pair-cat code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/paircat} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/oscillators/fock_state/paircat.yml.