Description
A concatenated qubit-into-\(n\)-mode code obtained by encoding each qubit of a quantum repetition code into a two-component cat code in its cat-state basis.
A basis of codewords for the two-component case, \begin{align} |\overline{\pm}\rangle\propto\left(\left|\alpha\right\rangle \pm\left|-\alpha\right\rangle \right)^{\otimes n} \tag*{(1)}\end{align} for any complex \(\alpha\).
Protection
The code can detect arbitrary losses in up to \(n/2\) modes. The cat-repetition code on a 2D mode lattice is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode [4].Gates
Fault-tolerant logical \(X\), CNOT, and Toffoli gates and a logical Hadamard synthesized from state preparation and measurement in the dual basis, without magic-state preparation or distillation [2].Using a physical bias-preserving CX between cat qubits, the logical \(\overline{\mathrm{CX}}\) gadget can be implemented transversally between repetition-code blocks; magic-state preparation using transversal \(ZZ(\theta)\) gates was also analyzed [3].Decoding
A measurement-code decoder for the repetition layer yields a \(\overline{\mathrm{CX}}\)-gadget threshold of about \(6\times 10^{-3}\) for \(n=5\); reaching a comparable threshold with naive repeated-syndrome decoding requires \(n=11\) and \(r=5\) [3].Fault Tolerance
Fault-tolerant logical \(X\), CNOT, and Toffoli gates and a logical Hadamard synthesized from state preparation and measurement in the dual basis, without magic-state preparation or distillation [2].Threshold
For dephasing bias \(\eta=10^4\), the cat-based logical \(\overline{\mathrm{CX}}\) gadget has threshold \(7.5\times 10^{-3}\), compared with \(3.55\times 10^{-3}\) for an earlier scheme; at \(\varepsilon=2.5\times 10^{-3}\), its circuit volume is about five times smaller [3].Realizations
Superconducting circuit devices: a repetition code out of two-component cat qubits has been realized for distances 3 and 5 [5].Cousins
- Quantum repetition code— The cat-repetition code is obtained by encoding each qubit of a quantum repetition code into a two-component cat code in its cat-state basis [2,3,6–8].
- Self-correcting quantum code— The cat-repetition code on a 2D mode lattice is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode [4].
- Coherent-state 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 \(n=1\).
Primary Hierarchy
Parents
The cat-repetition code is a concatenation whose outer code is the cat code in its cat-state basis.
Cat-repetition code
Children
The cat-repetition code for \(n=1\) reduces to the cat code.
References
- [1]
- J. Cohen and M. Mirrahimi, “Dissipation-induced continuous quantum error correction for superconducting circuits”, Physical Review A 90, (2014) arXiv:1409.6759 DOI
- [2]
- J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Physical Review X 9, (2019) arXiv:1904.09474 DOI
- [3]
- S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020) arXiv:1905.00450 DOI
- [4]
- S. Lieu, Y.-J. Liu, and A. V. Gorshkov, “Candidate for a passively protected quantum memory in two dimensions”, (2023) arXiv:2205.09767
- [5]
- H. Putterman et al., “Hardware-efficient quantum error correction via concatenated bosonic qubits”, Nature 638, 927 (2025) arXiv:2409.13025 DOI
- [6]
- J. Guillaud and M. Mirrahimi, “Error rates and resource overheads of repetition cat qubits”, Physical Review A 103, (2021) arXiv:2009.10756 DOI
- [7]
- C. Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes”, PRX Quantum 3, (2022) arXiv:2012.04108 DOI
- [8]
- F.-M. L. Régent, C. Berdou, Z. Leghtas, J. Guillaud, and M. Mirrahimi, “High-performance repetition cat code using fast noisy operations”, Quantum 7, 1198 (2023) arXiv:2212.11927 DOI
Page edit log
- Victor V. Albert (2024-12-06) — most recent
Cite as:
“Cat-repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cat_repetition