Description
A concatenated qubit-into-\(n\)-mode code (for odd \(n\)) whose inner code is a quantum repetition code and whose outer code is the two-component cat code in its coherent-state basis.
A basis of codewords is \begin{align} |\overline{\pm}\rangle\propto\left|\pm\alpha\right\rangle ^{\otimes n} \tag*{(1)}\end{align} for \(|\alpha| > 0\).
Protection
For odd \(n\), the code has \(d_X=1\) and minimum Euclidean distance \(d_Z = 4n\) [3], so it does not protect against losses but suppresses dephasing exponentially in \(n|\alpha|^2\).Encoding
Lindbladian-based dissipative encoding with dissipators \(\hat{a}_j \hat{a}_k - \alpha^2\) for neighboring modes \(j,k\) on a 1D line as well as local dissipators \(\hat{a}_j^2 - \alpha^2\). Encoding passively protects against cavity dephasing.Cousins
- Cat-repetition code— The cat (coherent-state) repetition code is a concatenation whose outer code is the (two-component) cat code in its cat (coherent-state) basis. For the two-component case, both reduce to the two-component cat code at \(n=1\).
- Quantum repetition code— Two-component cat codes in the coherent-state basis have been concatenated with quantum repetition codes [1,2].
Member of code lists
Primary Hierarchy
Parents
The coherent-state repetition code is a concatenation whose outer code is the cat code in its coherent-state basis.
For odd \(n\), the coherent-state repetition code is a tiger code whose matrix \(G\) is the cyclic repetition generator matrix over the integers and whose matrix \(H\) is zero [3]. For even \(n\), or after removing the last row to impose open boundaries, the construction yields a logical-rotor variant instead of a logical qubit.
Coherent-state repetition code
Children
The coherent-state repetition code for \(n=1\) reduces to the two-component cat code.
References
- [1]
- H. Jeong and M. S. Kim, “Efficient quantum computation using coherent states”, Physical Review A 65, (2002) arXiv:quant-ph/0109077 DOI
- [2]
- T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states”, Physical Review A 68, (2003) arXiv:quant-ph/0306004 DOI
- [3]
- Y. Xu, Y. Wang, C. Vuillot, and V. V. Albert, “Letting the Tiger out of Its Cage: Bosonic Coding without Concatenation”, Physical Review X 15, (2025) arXiv:2411.09668 DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-12-06)
Cite as:
“Coherent-state repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/coherent_state_repetition, arXiv:2606.11484