# Cat code[1]

## Description

Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator which utilizes a constellation of \(q(S+1)\) coherent states distributed equidistantly around a circle in phase space of radius \(\alpha\).

Codewords for a qubit code (\(q=2\)) consist of a coherent state \(|\alpha\rangle\) projected onto a subspace of Fock state number modulo \(2(S+1)\). The logical state \(|\overline{0}\rangle\) is in the \(\{|0\rangle , |2(S+1)\rangle , |4(S+1)\rangle \cdots \}\) Fock-state subspace, while \(|\overline{1}\rangle\) is in the \(\{|(S+1)\rangle, |3(S+1)\rangle , |5(S+1)\rangle , |7(S+1)\rangle \cdots \}\) subspace.

## Protection

## Encoding

## Gates

## Decoding

## Realizations

## Parents

- Bosonic rotation code — The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) [9].
- Coherent-state constellation code — Cat-code codewords are constructed using a coherent-state constellation that forms the cyclic group \(\mathbb{Z}_{2S+2}\).

## Child

- Two-component cat code — The cat code reduces to its two-component version for \(S=0\)

## Cousins

- Number-phase code — In the limit as \(N,S \to \infty\), phase measurement in the cat code has vanishing variance, just like in a number-phase code [9].
- Binomial code — For a fixed \(S\), binomial codes with \(N \to \infty\) coincide with cat codes as \(\alpha \to \infty\) [10].
- PSK c-q code — PSK c-q (cat) codes are used to transmit classical (quantum) information using (superpositions of) single-mode coherent states distributed on a circle over quantum channels.
- Pair-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.
- Phase-shift keyring (PSK) code — PSK (cat) codes are used to transmit classical (quantum) information using (superpositions of) single-mode coherent states distributed on a circle over classical (quantum) channels.

## References

- [1]
- Z. Leghtas et al., “Hardware-Efficient Autonomous Quantum Memory Protection”, Physical Review Letters 111, (2013). DOI; 1207.0679
- [2]
- M. Wolinsky and H. J. Carmichael, “Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions”, Physical Review Letters 60, 1836 (1988). DOI
- [3]
- L. Krippner, W. J. Munro, and M. D. Reid, “Transient macroscopic quantum superposition states in degenerate parametric oscillation: Calculations in the large-quantum-noise limit using the positive<i>P</i>representation”, Physical Review A 50, 4330 (1994). DOI
- [4]
- E. E. Hach III and C. C. Gerry, “Generation of mixtures of Schrödinger-cat states from a competitive two-photon process”, Physical Review A 49, 490 (1994). DOI
- [5]
- L. Gilles, B. M. Garraway, and P. L. Knight, “Generation of nonclassical light by dissipative two-photon processes”, Physical Review A 49, 2785 (1994). DOI
- [6]
- M. Mirrahimi et al., “Dynamically protected cat-qubits: a new paradigm for universal quantum computation”, New Journal of Physics 16, 045014 (2014). DOI; 1312.2017
- [7]
- V. V. Albert et al., “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016). DOI; 1503.00194
- [8]
- Nissim Ofek et al., “Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information”. 1602.04768
- [9]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071
- [10]
- M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016). DOI; 1602.00008

## Page edit log

- Victor V. Albert (2022-11-06) — most recent
- Victor V. Albert (2022-07-03)
- Alexander Grimm (2022-07-03)
- Victor V. Albert (2022-01-11)
- Joseph T. Iosue (2021-12-19)
- Yijia Xu (2021-12-14)

## Zoo code information

## Cite as:

“Cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/cat