# Two-component cat code[1]

## Description

Code whose codespace is spanned by two coherent states \(\left|\pm\alpha\right\rangle\) for nonzero complex \(\alpha\). An orthonormal basis for the codespace consists of the bosonic cat states \begin{align} |\overline{\pm}\rangle=\frac{\left|\alpha\right\rangle \pm\left|-\alpha\right\rangle }{\sqrt{2\left(1\pm e^{-2|\alpha|^{2}}\right)}} \tag*{(1)}\end{align} for any complex \(\alpha\).

A closely related approximate cat code is called T4C code [2].

## Protection

Two-legged cat codes for large \(\alpha\) provide protection against modal dephasing, i.e., diffusion of the angular degree of freedom of the mode. They do not protect against photon loss events, but there exist modifications based on sign alternation [3], squeezing [4–6], and detuning [7] that can add such protection.

## Encoding

Lindbladian-based dissipative encoding [8] utilizing two-photon absorption [9–12]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(|\alpha|^2\) [8].Hamiltonian-based 'Kerr-cat' encoding utilizing the Kerr-effect Hamiltonian [13] (see also Ref. [14]).Colored dissipation [15].Combined dissipative and Hamiltonian-based encoding utilizing two-photon exchange with an ancillary qubit [16].Critical encoding at non-zero detuning [17].

## Gates

Universal gates in the quantum optical setting can be performed using teleportation, Bell measurements, displacements, and rotations [18]. An earlier protocol requires a nonlinear interaction and uses state teleportation [19].Universal gates in the microwave setting can be performed using displacement operators and a rotation based on the Kerr nonlinearity [8]. Kerr nonlinearity converts coherent states into Yurke-Stoler states [20].Bias-preserving Hamiltonian-based CNOT gate utilizing an \(X\) gate with a topological Berry phase [21,22]. Bias-preserving SWAP gate [23].

## Decoding

All-optical decoder [24] based on Knill error correction (a.k.a. telecorrection [25]), which is based on teleportation [26,27].

## Fault Tolerance

Fault-tolerant error-correction procedure using small amplitude coherent states [28].Bias-preserving Hamiltonian-based CNOT gate [22] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [21,22].Ancilla qubits encoded in two-component cat codes yield fault-tolerant syndrome extraction circuits [29].

## Realizations

Lindbladian-based [30,31] and Hamiltonian-based 'Kerr-cat' [32] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [31] 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 up to a bit-flip time of 1ms [33] and later 10 seconds [34], with the latter scheme repurposing a holonomic gate [35] as a measurement.T4C code realized in a superconducting circuit device by the Wang group [2].

## Notes

Pedagogical introduction to cat codes in the context of microwave cavities can be found in Refs. [36,37], and in the context of optical systems in books [38–40].

## Parent

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

## Cousins

- Hamiltonian-based code — The two-legged cat code forms the ground-state subspace of a Kerr Hamiltonian [13].
- Quantum repetition code — Two-legged cat and quantum repetition codes can be thought of as classical codes because they protect against only one type of noise. Two-legged cat codes (quantum repetition) codes suppress cavity dephasing (bit-flip) noise exponentially with \(|\alpha|^2\) (\(n\)). The stability offered by cat codes has been linked to several favorable properties of phases of matter associated with the repetition-code Hamiltonian [41,42].
- Coherent-state c-q code — Two-component cat codes can be thought of as coherent-state c-q codes because they protect against only one type of noise and thus only reliably store classical information.
- Self-correcting quantum code — A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode [43].
- Binary PSK (BPSK) code — BPSK (two-component cat) codes are used to transmit classical (quantum) information using (superpositions of) antipodal coherent states over classical (quantum) channels.
- BPSK c-q code — BPSK c-q (two-component cat) codes are used to transmit classical (quantum) information using (superpositions of) antipodal coherent states over quantum channels.
- Quantum spherical code (QSC) — CSS codes concatenated with two-component cat codes form QSCs which have a weight-based notion of distance.
- Spin cat code — The Spin-cat code construction utilizes the Holstein-Primakoff mapping [44] (see also [45]) to convert cat codes into codes for spin systems.

## 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) arXiv:quant-ph/9809037 DOI
- [2]
- J. M. Gertler et al., “Protecting a bosonic qubit with autonomous quantum error correction”, Nature 590, 243 (2021) arXiv:2004.09322 DOI
- [3]
- L. Li et al., “Phase-engineered bosonic quantum codes”, Physical Review A 103, (2021) arXiv:1901.05358 DOI
- [4]
- D. S. Schlegel, F. Minganti, and V. Savona, “Quantum error correction using squeezed Schrödinger cat states”, Physical Review A 106, (2022) arXiv:2201.02570 DOI
- [5]
- Q. Xu et al., “Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits”, (2022) arXiv:2210.13406
- [6]
- T. Hillmann and F. Quijandría, “Quantum error correction with dissipatively stabilized squeezed-cat qubits”, Physical Review A 107, (2023) arXiv:2210.13359 DOI
- [7]
- D. Ruiz et al., “Two-photon driven Kerr quantum oscillator with multiple spectral degeneracies”, Physical Review A 107, (2023) arXiv:2211.03689 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]
- 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
- [10]
- 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
- [11]
- 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
- [12]
- 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
- [13]
- 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) arXiv:1605.09408 DOI
- [14]
- H. Goto, “Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network”, Scientific Reports 6, (2016) arXiv:1510.02566 DOI
- [15]
- H. Putterman et al., “Stabilizing a Bosonic Qubit Using Colored Dissipation”, Physical Review Letters 128, (2022) arXiv:2107.09198 DOI
- [16]
- R. Gautier, A. Sarlette, and M. Mirrahimi, “Combined Dissipative and Hamiltonian Confinement of Cat Qubits”, PRX Quantum 3, (2022) arXiv:2112.05545 DOI
- [17]
- L. Gravina, F. Minganti, and V. Savona, “Critical Schrödinger Cat Qubit”, PRX Quantum 4, (2023) arXiv:2208.04928 DOI
- [18]
- T. C. Ralph et al., “Quantum computation with optical coherent states”, Physical Review A 68, (2003) arXiv:quant-ph/0306004 DOI
- [19]
- H. Jeong and M. S. Kim, “Efficient quantum computation using coherent states”, Physical Review A 65, (2002) arXiv:quant-ph/0109077 DOI
- [20]
- B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion”, Physical Review Letters 57, 13 (1986) DOI
- [21]
- J. Guillaud and M. Mirrahimi, “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Physical Review X 9, (2019) arXiv:1904.09474 DOI
- [22]
- S. Puri et al., “Bias-preserving gates with stabilized cat qubits”, Science Advances 6, (2020) arXiv:1905.00450 DOI
- [23]
- J. Guillaud and M. Mirrahimi, “Error rates and resource overheads of repetition cat qubits”, Physical Review A 103, (2021) arXiv:2009.10756 DOI
- [24]
- J. Hastrup and U. L. Andersen, “All-optical cat-code quantum error correction”, (2021) arXiv:2108.12225
- [25]
- C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
- [26]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [27]
- E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
- [28]
- A. P. Lund, T. C. Ralph, and H. L. Haselgrove, “Fault-Tolerant Linear Optical Quantum Computing with Small-Amplitude Coherent States”, Physical Review Letters 100, (2008) arXiv:0707.0327 DOI
- [29]
- S. Puri et al., “Stabilized Cat in a Driven Nonlinear Cavity: A Fault-Tolerant Error Syndrome Detector”, Physical Review X 9, (2019) arXiv:1807.09334 DOI
- [30]
- Z. Leghtas et al., “Confining the state of light to a quantum manifold by engineered two-photon loss”, Science 347, 853 (2015) arXiv:1412.4633 DOI
- [31]
- S. Touzard et al., “Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation”, Physical Review X 8, (2018) arXiv:1705.02401 DOI
- [32]
- A. Grimm et al., “Stabilization and operation of a Kerr-cat qubit”, Nature 584, 205 (2020) arXiv:1907.12131 DOI
- [33]
- R. Lescanne et al., “Exponential suppression of bit-flips in a qubit encoded in an oscillator”, Nature Physics 16, 509 (2020) arXiv:1907.11729 DOI
- [34]
- U. Réglade et al., “Quantum control of a cat-qubit with bit-flip times exceeding ten seconds”, (2023) arXiv:2307.06617
- [35]
- V. V. Albert et al., “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016) arXiv:1503.00194 DOI
- [36]
- J. Guillaud, J. Cohen, and M. Mirrahimi, “Quantum computation with cat qubits”, SciPost Physics Lecture Notes (2023) arXiv:2203.03222 DOI
- [37]
- Shruti Puri, QEC when the noise is biased, 2019.
- [38]
- S. Haroche and J.-M. Raimond, Exploring the Quantum (Oxford University Press, 2006) DOI
- [39]
- H. Jeong and T. C. Ralph, “Schrödinger Cat States for Quantum Information Processing”, Quantum Information with Continuous Variables of Atoms and Light 159 (2007) DOI
- [40]
- H. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley, 2019) DOI
- [41]
- F. Minganti et al., “Spectral theory of Liouvillians for dissipative phase transitions”, Physical Review A 98, (2018) arXiv:1804.11293 DOI
- [42]
- S. Lieu et al., “Symmetry Breaking and Error Correction in Open Quantum Systems”, Physical Review Letters 125, (2020) arXiv:2008.02816 DOI
- [43]
- S. Lieu, Y.-J. Liu, and A. V. Gorshkov, “Candidate for a passively protected quantum memory in two dimensions”, (2023) arXiv:2205.09767
- [44]
- T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
- [45]
- C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI

## Page edit log

- Victor V. Albert (2022-11-06) — most recent
- Hyunseok Jeong (2022-11-06)

## Cite as:

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