Pair-cat code

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-component case, \(|\overline{0}_{\gamma,\Delta}\rangle \sim (|\gamma_\Delta \rangle + (-1)^\Delta |i\gamma_\Delta\rangle)/\sqrt{2}\) and \(|\overline{1}_{\gamma,\Delta}\rangle \sim (|\gamma_\Delta\rangle - (-1)^\Delta |i \gamma\rangle)/\sqrt{2}\), 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 [2–5] with complex amplitude \(\alpha\), up to normalization. The asymptotic expression of the codewords is valid in the limit of large energy, \(|\alpha|^2\to\infty\).

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-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.

Gates

Hamiltonian \(X\), \(XX\), \(Z\) gates, holonomic \(Z\) gate, control-phase gate.Bias-preserving gates [6].

Decoding

Lindbladian-based dissipative encoding utilizing two-mode two-photon absorption [4]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(\gamma^2\).

Realizations

Microwave cavities coupled to superconducting circuits by the Wang group [7].

Parents

Fock-state bosonic code
Tiger code — The pair-cat code is a tiger code with \(G = (2,2)\) and \(H = (1,-1)\) [8].

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, S. O. Mundhada, A. Grimm, S. Touzard, M. H. Devoret, and L. Jiang, “Pair-cat codes: autonomous error-correction with low-order nonlinearity”, Quantum Science and Technology 4, 035007 (2019) arXiv:1801.05897 DOI
[2]: D. Bhaumik, K. Bhaumik, and B. Dutta-Roy, “Charged bosons and the coherent state”, Journal of Physics A: Mathematical and General 9, 1507 (1976) DOI
[3]: A. O. Barut and L. Girardello, “New “Coherent” States associated with non-compact groups”, Communications in Mathematical Physics 21, 41 (1971) DOI
[4]: 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
[5]: G. S. Agarwal, “Nonclassical statistics of fields in pair coherent states”, Journal of the Optical Society of America B 5, 1940 (1988) DOI
[6]: 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
[7]: J. M. Gertler, S. van Geldern, S. Shirol, L. Jiang, and C. Wang, “Experimental Realization and Characterization of Stabilized Pair Coherent States”, (2022) arXiv:2209.11643
[8]: Y. Xu, Y. Wang, C. Vuillot, and V. V. Albert, “Letting the tiger out of its cage: bosonic coding without concatenation”, (2024) arXiv:2411.09668

Page edit log

Victor V. Albert (2022-08-16) — most recent
Yijia Xu (2022-05-03)

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/edit/main/codes/quantum/oscillators/fock_state/paircat.yml.