# 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. These projected coherent states make up generalized cat states [2,3].

## Protection

## Encoding

## Gates

## Decoding

## Fault Tolerance

## Realizations

## Parents

- Quantum spherical code (QSC) — Cat codes are QSCs on the one-dimensional complex sphere.
- Bosonic rotation code — The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) [16].
- Group-representation code — Cat codes are group representation codes with \(G\) being a cyclic group [17].

## 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 [16].
- Phase-shift keying (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.
- Polygon code — The \(q(S+1)\)-component cat coherent-state constellation forms the vertices of a \(q(S+1)\)-gon.
- 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.
- Binomial code — For a fixed \(S\), binomial codes with \(N \to \infty\) coincide with cat codes as \(\alpha \to \infty\) [18].
- Hybrid cat code — Hybrid cat codewords consist of a bosonic mode in either coherent or cat states.

## References

- [1]
- Z. Leghtas et al., “Hardware-Efficient Autonomous Quantum Memory Protection”, Physical Review Letters 111, (2013) arXiv:1207.0679 DOI
- [2]
- V. V. Dodonov, I. A. Malkin, and V. I. Man’ko, “Even and odd coherent states and excitations of a singular oscillator”, Physica 72, 597 (1974) DOI
- [3]
- O. Castaños, R. López-Peña, and V. I. Man’ko, “Crystallized schrödinger cat states”, Journal of Russian Laser Research 16, 477 (1995) DOI
- [4]
- 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
- [5]
- 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 positivePrepresentation”, Physical Review A 50, 4330 (1994) DOI
- [6]
- 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
- [7]
- 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
- [8]
- M. Mirrahimi et al., “Dynamically protected cat-qubits: a new paradigm for universal quantum computation”, New Journal of Physics 16, 045014 (2014) arXiv:1312.2017 DOI
- [9]
- D. Su, C. R. Myers, and K. K. Sabapathy, “Conversion of Gaussian states to non-Gaussian states using photon-number-resolving detectors”, Physical Review A 100, (2019) arXiv:1902.02323 DOI
- [10]
- V. V. Albert et al., “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016) arXiv:1503.00194 DOI
- [11]
- T. Tsunoda et al., “Error-detectable bosonic entangling gates with a noisy ancilla”, (2022) arXiv:2212.11196
- [12]
- Q. Xu et al., “Fault-Tolerant Operation of Bosonic Qubits with Discrete-Variable Ancillae”, (2023) arXiv:2310.20578
- [13]
- L. Sun et al., “Tracking photon jumps with repeated quantum non-demolition parity measurements”, Nature 511, 444 (2014) arXiv:1311.2534 DOI
- [14]
- N. Ofek et al., “Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information”, (2016) arXiv:1602.04768
- [15]
- S. Rosenblum et al., “Fault-tolerant detection of a quantum error”, Science 361, 266 (2018) arXiv:1803.00102 DOI
- [16]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020) arXiv:1901.08071 DOI
- [17]
- A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621
- [18]
- M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016) arXiv:1602.00008 DOI

## 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)

## Cite as:

“Cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/cat