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 \(|\overline{1}_{\gamma,\Delta}\rangle \sim (|\gamma_\Delta\rangle - (-1)^\Delta |i \gamma\rangle)/\sqrt{2}\) [1,2], 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 [3–6] 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 [7].Decoding
Lindbladian-based dissipative encoding utilizing two-mode two-photon absorption [5]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(\gamma^2\).Realizations
Microwave cavities coupled to superconducting circuits by the Wang group [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.
Primary Hierarchy
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]
- C. C. Gerry and R. Grobe, “Nonclassical properties of correlated two-mode Schrödinger cat states”, Physical Review A 51, 1698 (1995) DOI
- [3]
- 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
- [4]
- A. O. Barut and L. Girardello, “New “Coherent” States associated with non-compact groups”, Communications in Mathematical Physics 21, 41 (1971) DOI
- [5]
- 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
- [6]
- G. S. Agarwal, “Nonclassical statistics of fields in pair coherent states”, Journal of the Optical Society of America B 5, 1940 (1988) DOI
- [7]
- 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
- [8]
- 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
- [9]
- 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