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

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.

Encoding

Lindbladian-based dissipative encoding utilizing multi-photon generalization of two-photon absorption [2][3][4][5]. Encoding passively protects against modal dephasing, suppressing dephasing noise exponentially with \(|\alpha|^2\) [6].

Gates

Holonomic gates utilizing the Berry phase of coherent states are universal [7].Universal gates in the microwave setting for the \(S=1\) code can be performed using squeezing operators and a rotation based on the Kerr nonlinearity [6].

Decoding

Measuring the Fock-state number modulo \(2S\) can be used to determine if photon loss or excitation errors occurred.

Realizations

Four-legged (\(S=1\)) cat code has been realized in a superconducting circuit device [8]. This paper is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the cavity lifetime.

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

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
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Zoo code information

Internal code ID: cat

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Zoo Code ID: cat

Cite as:
“Cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/cat
BibTeX:
@incollection{eczoo_cat, title={Cat code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/cat} }
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https://errorcorrectionzoo.org/c/cat

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/oscillators/constellations/cat.yml.