## Description

Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator. 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. An alternative basis, valid for for general \(q\) and \(\alpha\neq 0\), consists of \(q\) coherent states distributed equidistanctly around a circle in phase space of radius \(\alpha\).

## Protection

Due to the spacing between sets of Fock states, the distance between two distinct logical states is \(d=S+1\). Hence, this code is able to detect \(S\)-photon loss error. Two-legged cat codes (\(S=0\)) do not protect against loss events, but there exist modifications based on sign alternation [3] or squeezing [4] that add such protection.

## Encoding

Lindbladian-based dissipative encoding [5] utilizing multi-photon absorption [6][7][8][9]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(\alpha^2\) [5].Hamiltonian-based 'Kerr-cat' encoding utilizing the Kerr effect [10] (see also Ref. [11]).Combined dissipative and Hamiltonian-based encoding utilizing two-photon exchange for \(S=0\) codes [12].

## Gates

For \(S=0\), universal gates in the quantum optical setting can be performed using teleportation, Bell measurements, displacements, and rotations [13].For \(S=0\), universal gates in the microwave setting can be performed using displacement operators and a rotation based on the Kerr nonlinearity [5]. For \(S=1\), squeezing replaces displacements.Holonomic gates utilizing the Berry phase of coherent states are universal [14].Bias-preserving Hamiltonian-based CNOT gate utilizing an \(X\) gate with a topological Berry phase [15][16].

## Decoding

Measurement in the Fock basis. For a \(2(S+1)\) cat code, a number measurement returns outcome \(2(S+1)k\), if \(k\) is even, then it corresponds to logical 0 state; if \(k\) is odd, then it corresponds to logical 1 state.

## Fault Tolerance

Bias-preserving Hamiltonian-based CNOT gate [16] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [15][16].Ancilla qubits encoded in cat codes yield fault-tolerant syndrome extraction circuits [17].

## Realizations

Two-legged (\(S=0\)) Lindbladian-based [18][19] and Hamiltonian-based 'Kerr-cat' encoding [20] has been achieved in superconducting circuit devices by the Devoret group; Ref. [19] also demonstrated a displacement-based gate. The Lindbladian-based scheme has further achieved a suppression of bit-flip errors that is exponential in the average photon number [21] up to a bit-flip time of 1ms. A bit-flip time of 1s has been achieved in a similar system in the classical bit regime [22].Four-legged (\(S=1\)) cat code has been realized in a superconducting circuit device [23]. This paper is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the lifetime of the noisiest constituent of the system.Approximate version of the \(S=0\) code realized in a superconducting circuit device by the Wang group [24].

## Notes

Pedagogical introduction to cat codes in the context of microwave cavities can be found in Refs. [25][26], and in the context of optical systems in books [27][28].

## Parent

- Bosonic rotation code — The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) [29].

## 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 [29].
- Hamiltonian-based code — Two-legged cat codewords form ground-state subspace of a Kerr Hamiltonian [10].
- Binomial code — For a fixed \(S\), binomial codes with \(N \to \infty\) coincide with cat codes as \(\alpha \to \infty\) [30].
- 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.

## References

- [1]
- P. T. Cochrane, G. J. Milburn, and W. J. Munro, “Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping”, Physical Review A 59, 2631 (1999). DOI; quant-ph/9809037
- [2]
- Z. Leghtas et al., “Hardware-Efficient Autonomous Quantum Memory Protection”, Physical Review Letters 111, (2013). DOI; 1207.0679
- [3]
- L. Li et al., “Phase-engineered bosonic quantum codes”, Physical Review A 103, (2021). DOI; 1901.05358
- [4]
- David S. Schlegel, Fabrizio Minganti, and Vincenzo Savona, “Quantum error correction using squeezed Schrödinger cat states”. 2201.02570
- [5]
- 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
- [6]
- 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
- [7]
- 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
- [8]
- 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
- [9]
- 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
- [10]
- S. Puri, S. Boutin, and A. Blais, “Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving”, npj Quantum Information 3, (2017). DOI; 1605.09408
- [11]
- H. Goto, “Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network”, Scientific Reports 6, (2016). DOI; 1510.02566
- [12]
- Ronan Gautier, Alain Sarlette, and Mazyar Mirrahimi, “Combined Dissipative and Hamiltonian Confinement of Cat Qubits”. 2112.05545
- [13]
- T. C. Ralph et al., “Quantum computation with optical coherent states”, Physical Review A 68, (2003). DOI; quant-ph/0306004
- [14]
- V. V. Albert et al., “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016). DOI; 1503.00194
- [15]
- J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Physical Review X 9, (2019). DOI; 1904.09474
- [16]
- S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020). DOI; 1905.00450
- [17]
- S. Puri et al., “Stabilized Cat in a Driven Nonlinear Cavity: A Fault-Tolerant Error Syndrome Detector”, Physical Review X 9, (2019). DOI; 1807.09334
- [18]
- Z. Leghtas et al., “Confining the state of light to a quantum manifold by engineered two-photon loss”, Science 347, 853 (2015). DOI; 1412.4633
- [19]
- S. Touzard et al., “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation”, Physical Review X 8, (2018). DOI; 1705.02401
- [20]
- A. Grimm et al., “Stabilization and operation of a Kerr-cat qubit”, Nature 584, 205 (2020). DOI; 1907.12131
- [21]
- R. Lescanne et al., “Exponential suppression of bit-flips in a qubit encoded in an oscillator”, Nature Physics 16, 509 (2020). DOI; 1907.11729
- [22]
- C. Berdou et al., “One hundred second bit-flip time in a two-photon dissipative oscillator”. 2204.09128
- [23]
- Nissim Ofek et al., “Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information”. 1602.04768
- [24]
- J. M. Gertler et al., “Protecting a bosonic qubit with autonomous quantum error correction”, Nature 590, 243 (2021). DOI; 2004.09322
- [25]
- Jérémie Guillaud, Joachim Cohen, and Mazyar Mirrahimi, “Quantum computation with cat qubits”. 2203.03222
- [26]
- Shruti Puri, QEC when the noise is biased, 2019.
- [27]
- S. Haroche and J.-M. Raimond, Exploring the Quantum (Oxford University Press, 2006). DOI
- [28]
- H. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley, 2019). DOI
- [29]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071
- [30]
- 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 (2022-07-03) — most recent
- 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.), 2022. https://errorcorrectionzoo.org/c/cat