## 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 (\(S=1\)) 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 utilizing multi-photon absorption [5].Hamiltonian-based 'Kerr-cat' encoding utilizing the Kerr effect [6] (see also Ref. [7]).Combined dissipative and Hamiltonian-based encoding utilizing two-photon exchange for \(S=1\) codes [8].

## Gates

For \(S=1\), universal gates can be performed using displacement operators and a rotation based on the Kerr nonlinearity [5]. For \(S=2\), squeezing replaces displacements.Holonomic gates utilizing the Berry phase of coherent states are universal [9].Bias-preserving CNOT gate utilizing a topological Berry phase [10].

## 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 CNOT gate [10] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [11][10].

## Realizations

Two-legged (\(S=1\)) cat code has been realized in a superconducting circuit device by the Devoret group [12]. Exponential suppression of bit-flip errors achieved [13] up to a bit-flip time of 1 ms. A bit-flip time of up to 1 sec has been achieved while away from the logical-qubit regime [14].Four-legged (\(S=2\)) cat code has been realized in a superconducting circuit device [15]. 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.

## Notes

Pedagogical introduction to cat codes can be found in Ref. [16].

## Parent

- Bosonic rotation code — The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) [17].

## 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 [17].
- Hamiltonian-based code — Two-legged cat codewords form ground-state subspace of a Kerr Hamiltonian [6].
- Binomial code — For a fixed \(S\), binomial codes with \(N \to \infty\) coincide with cat codes as \(\alpha \to \infty\) [18].

## Zoo code information

## References

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- [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
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- David S. Schlegel, Fabrizio Minganti, and Vincenzo Savona, “Quantum error correction using squeezed Schrödinger cat states”. 2201.02570
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- 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
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- 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
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- H. Goto, “Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network”, Scientific Reports 6, (2016). DOI; 1510.02566
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- Ronan Gautier, Alain Sarlette, and Mazyar Mirrahimi, “Combined Dissipative and Hamiltonian Confinement of Cat Qubits”. 2112.05545
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- V. V. Albert et al., “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016). DOI; 1503.00194
- [10]
- S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020). DOI; 1905.00450
- [11]
- J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Physical Review X 9, (2019). DOI; 1904.09474
- [12]
- 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
- [13]
- R. Lescanne et al., “Exponential suppression of bit-flips in a qubit encoded in an oscillator”, Nature Physics 16, 509 (2020). DOI; 1907.11729
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- C. Berdou et al., “One hundred second bit-flip time in a two-photon dissipative oscillator”. 2204.09128
- [15]
- N. Ofek et al., “Extending the lifetime of a quantum bit with error correction in superconducting circuits”, Nature 536, 441 (2016). DOI
- [16]
- Jérémie Guillaud, Joachim Cohen, and Mazyar Mirrahimi, “Quantum computation with cat qubits”. 2203.03222
- [17]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071
- [18]
- M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016). DOI; 1602.00008

## Cite as:

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