# Concatenated cat code[1]

## Description

A concatenated code whose outer code is a cat code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another cat outer code. Most examples encode physical qubits of an inner stabilizer code into the two-component cat code.

## Protection

The cat code suppresses dephasing errors exponentially with the size of its coherent states, so the inner code (e.g., a quantum repetition code [2–5]) can be highly biased toward one type of noise while still ensuring good performance.

A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode [6].

## Realizations

Superconducting circuit devices: a repetition code out of two-component cat qubits has been realized for distances 3 and 5 [7].

## Parent

## Cousins

- Quantum repetition code — Two-component cat codes have been concatenated with quantum repetition codes [2–5,8].
- Rotated surface code — Cat codes have been concatenated with rotated surface codes [8].
- Low-density parity-check (LDPC) code — Cat codes have been concatenated with LDPC codes (treated as qubit stabilizer codes) [9].
- Lechner-Hauke-Zoller (LHZ) code — LHZ parity-codes have been concatenated with cat codes [10].
- \([[7,1,3]]\) Steane code — Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) [11].
- Quantum Golay code — Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) [11].
- Self-correcting quantum code — A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode [6].
- Quantum spherical code (QSC) — CSS codes concatenated with two-component cat codes form QSCs which have a weight-based notion of distance.

## 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]
- J. Guillaud and M. Mirrahimi, “Error rates and resource overheads of repetition cat qubits”, Physical Review A 103, (2021) arXiv:2009.10756 DOI
- [5]
- F.-M. L. Régent et al., “High-performance repetition cat code using fast noisy operations”, Quantum 7, 1198 (2023) arXiv:2212.11927 DOI
- [6]
- S. Lieu, Y.-J. Liu, and A. V. Gorshkov, “Candidate for a passively protected quantum memory in two dimensions”, (2023) arXiv:2205.09767
- [7]
- H. Putterman et al., “Hardware-efficient quantum error correction using concatenated bosonic qubits”, (2024) arXiv:2409.13025
- [8]
- C. Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes”, PRX Quantum 3, (2022) arXiv:2012.04108 DOI
- [9]
- D. Ruiz et al., “LDPC-cat codes for low-overhead quantum computing in 2D”, (2024) arXiv:2401.09541
- [10]
- A. Messinger et al., “Fault-tolerant quantum computing with the parity code and noise-biased qubits”, (2024) arXiv:2404.11332
- [11]
- A. P. Lund, T. C. Ralph, and H. L. Haselgrove, “Fault-Tolerant Linear Optical Quantum Computing with Small-Amplitude Coherent States”, Physical Review Letters 100, (2008) arXiv:0707.0327 DOI

## Page edit log

- Victor V. Albert (2024-07-17) — most recent

## Cite as:

“Concatenated cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cat_concatenated