# 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 non-linear 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. A bit-flip time of 1s has been achieved in a similar system in the classical bit regime [36].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. [37,38], and in the context of optical systems in books [39–41].

## 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 [42,43].
- 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 [44].
- 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 [45] (see also [46]) to convert cat codes into codes for spin systems.

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## 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