## Description

## Protection

Amplitude damping noise: The amplitude damping (AD) channel is a bosonic channel that models loss of particles in a bosonic mode (a.k.a. photon loss, pure loss, or fiber attenuation). Its Kraus operators are proportional to powers of a mode's annihilation operator \(a\), with the power signifying the number of particles lost during the error, \begin{align} E_{\ell}=\left(\frac{\gamma}{1-\gamma}\right)^{\ell/2}\frac{a^{\ell}}{\sqrt{\ell!}}\left(1-\gamma\right)^{\hat{n}/2}\,, \tag*{(1)}\end{align} where \(\gamma\in[0,1)\) is the noise rate [3,4]. For multiple modes, error set elements are tensor products of elements of the single-mode error set.

Restricting the channel to the first two Fock states \(\{|0\rangle,|1\rangle\}\) yields the non-Pauli qubit AD channel, which requires protecting against the loss error \(E_1\propto X+iY\) (instead of \(X\) and \(Y\) Pauli errors individually). Both channels are called AD since the context makes clear which one is being referred to. Other extension to qudits are also known [5].

Protection against AD noise is typically approximate because the tensor product of Kraus operators with all \(\ell=0\) is typically corrected only up to some order in \(\gamma\) [1,2]. For example, a qubit code that corrects a single AD error is one for which all tensor products \(E_{\ell_1}\otimes\cdots\otimes E_{\ell_n}\) with \(\ell_1+\cdots + \ell_n \leq 1\) are correctable (per the Knill-Laflamme conditions) up to order \(O(\gamma^2)\).

Certain codes also have intrinsic protection against AD, such as constant-excitation codes (CE), QSCs, or self-complementary codes. Amplitude damping can be thought of as a quantum analogue to asymmetric noise [6].

## Rate

## Parents

- Block quantum code
- Approximate quantum error-correcting code (AQECC) — Protection against AD noise is typically approximate because the tensor product of Kraus operators with all \(\ell=0\) is typically corrected only up to some order in \(\gamma\) [1,2].

## Children

- Quantum spherical code (QSC) — QSC codewords are superpositions of coherent states with the same energy, but coherent states are not eigenstates of the energy Hamiltonian. The AD Kraus operator \(E_{0}^{\otimes n}\) acts identically on each coherent state by shrinking the radius of the QSC's sphere.
- Fock-state bosonic code — Fock-state codes are designed to protect against bosonic AD noise.
- Constant-excitation (CE) code — Fock-state (and qubit) CE codes exactly protect against the AD Kraus operator \(E_{0}^{\otimes n}\) because it acts identically on all Fock (and qubit) states with the same excitation number [1,2].
- Jump code — Jump codes are designed to protect against qubit AD noise.
- Self-complementary quantum code — Self-complementary quantum codes consisting of computational basis states whose bitstrings are sufficiently spaced apart correct at least one AD error [12; Thm. 2.5][13; Thm. 2].
- Qudit GNU PI code — Qudit GNU PI codes protect against AD errors.

## Cousins

- Qubit CSS code — An \([[n,k,d_Z=t+1,d_X=2t+1]]\) qubit CSS code protects against \(t\) AD errors [15][14; Sec. 8.7].
- Concatenated qubit code — Concatenated quantum codes can protect against qubit AD [16].
- Dual-rail quantum code — Dual-rail qubits can be used to convert leakage and AD noise into erasure noise [17,18]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [19] that protects against \(d-1\) AD errors [15].
- Gottesman-Kitaev-Preskill (GKP) code — Transmission schemes with multimode GKP codes achieve, up to a constant-factor offset, the capacity of AD, displacement-noise, and thermal-noise Gaussian loss channels [20–23]. Combining AD noise with amplification yields displacement noise, the noise that GKP codes are designed to correct [23].
- \([[4,2,2]]\) Four-qubit code — The \([[4,1,2]]\) subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) (approximately) corrects a single AD error [1].
- Quantum parity code (QPC) — An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [19,24,25]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [15].
- Qubit stabilizer code — Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [19] that protects against \(d-1\) AD errors [15].

## References

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## Page edit log

- Victor V. Albert (2024-07-14) — most recent

## Cite as:

“Amplitude-damping (AD) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/ampdamp