Two-component cat code

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 [2] \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 [3].

Protection

Two-component cat codes for large \(\alpha\) provide protection against modal dephasing, i.e., diffusion of the angular degree of freedom of the mode. A single photon loss event maps the even and odd cat states approximately into each other and therefore acts as a logical bit flip rather than being corrected by the code. There exist modifications based on sign alternation [4], squeezing (yielding squeezed cat codes) [5–7], detuning [8], and addition of higher-order nonlinearities [9] that can add such protection.

Encoding

Lindbladian-based dissipative encoding and autonomous QEC [10] utilizing two-photon absorption [11–15]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(|\alpha|^2\) [10]. See Refs. [16,17] for analyses using displaced Fock states [18,19]. The Keldysh formalism yields non-perturbative bit-flip rates under various types of noise [20].Hamiltonian-based ‘Kerr-cat’ encoding utilizing the Kerr-effect Hamiltonian [21] (see also Ref. [22]).Colored dissipation [23].Combined dissipative and Hamiltonian-based encoding utilizing two-photon exchange with an ancillary qubit [24].Critical encoding at nonzero detuning [25].

Gates

Universal gates in the quantum optical setting can be performed using teleportation, Bell measurements, displacements, and rotations [26]. An earlier protocol requires a nonlinear interaction and uses state teleportation [27].Universal gates in the microwave setting can be performed using displacement operators and a rotation based on the Kerr nonlinearity [10]. Kerr nonlinearity converts coherent states into Yurke-Stoler states [28].Cat codes admit a noise bias-preserving Hamiltonian-based CNOT gate, utilizing an \(X\) gate with a topological Berry phase [29,30], and a bias-preserving SWAP gate [31]. The CNOT gate is part of a universal noise-bias-preserving gate set that can be made fault tolerant using concatenation [29,30].Cat-transmon entangling gate using an ancillary qubit [32].

Decoding

All-optical decoder [33] based on Knill error correction (a.k.a. telecorrection [34]), which is based on teleportation [35,36].

Fault Tolerance

Fault-tolerant error-correction procedure using small amplitude coherent states [37].Cat codes admit a noise bias-preserving Hamiltonian-based CNOT gate, utilizing an \(X\) gate with a topological Berry phase [29,30], and a bias-preserving SWAP gate [31]. The CNOT gate is part of a universal noise-bias-preserving gate set that can be made fault tolerant using concatenation [29,30].

Realizations

Lindbladian-based dissipative [38,39] and Hamiltonian-based ‘Kerr-cat’ [40] encodings have been achieved in superconducting circuit devices by the Devoret group; Ref. [39] 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 [41]. A bit-flip time of up to 10s has been achieved for the two-component cat code in the classical-bit regime [42–44]. A holonomic gate has been repurposed as a logical measurement [45]. The ‘Kerr-cat’ encoding and a \(\pi/2\) gate have been realized with the help of a band-block filter, yielding a bit-flip lifetime of 1 ms in the 10-photon regime [46] (see also Ref. [47]). Lindblad-based encoding achieved in a 2D cavity by AWS [48].T4C code realized in a superconducting circuit device by the Wang group [3].

Notes

Pedagogical introduction to cat codes in the context of microwave cavities can be found in Refs. [49,50], and in the context of optical systems in books [51–53].Ground states of the fluxonium superconducting qubit resemble two-component cat codewords [54].

Cousins

Hamiltonian-based code— The two-component cat code forms the ground-state subspace of a Kerr Hamiltonian [21].
Quantum repetition code— Two-component cat and quantum repetition codes can be thought of as classical codes because they protect against only one type of noise. Two-component 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 [55,56].
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.
Asymmetric quantum code— Cat codes admit a noise bias-preserving Hamiltonian-based CNOT gate, utilizing an \(X\) gate with a topological Berry phase [29,30], and a bias-preserving SWAP gate [31]. The CNOT gate is part of a universal noise-bias-preserving gate set that can be made fault tolerant using concatenation [29,30].
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.
Qubit stabilizer code— Ancilla modes can be used for syndrome extraction instead of ancilla qubits [57], and using two-component cat codes [58] yields fault-tolerant syndrome extraction circuits.
Spin cat code— The spin-cat code construction utilizes the Holstein-Primakoff mapping [59–61] to convert cat codes into codes for spin systems.

Member of code lists

Primary Hierarchy

Quantum Domain

Bosonic Kingdom

Bosonic codeQECC Quantum

Qudit-into-oscillator code

Fock-state bosonic codeAmplitude-damping (AD) QECC AQECC Quantum

Bosonic rotation codeMonolithic quantum QECC Quantum

Cat codeConcatenated quantum Amplitude-damping (AD) QECC AQECC Quantum

Parents

Cat code

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

Coherent-state repetition codeConcatenated quantum Amplitude-damping (AD) QECC AQECC Quantum

The coherent-state repetition code for \(n=1\) reduces to the two-component cat code.

Squeezed cat codeAmplitude-damping (AD) Monolithic quantum QECC AQECC Quantum

The squeezed cat code reduces to the two-component cat code when there is no squeezing.