## 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

- Squeezed cat code — Squeezing of coherent states allows for approximate protection against a single photon loss.
- 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.
- Squeezed fock-state code — The squeezed Fock-state code approximately protects against loss and dephasing errors, becoming exact in the \(r\to\infty\) limit.
- 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].
- Post-selected PI code
- 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

- [1]
- D. W. Leung et al., “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997) arXiv:quant-ph/9704002 DOI
- [2]
- I. L. Chuang, D. W. Leung, and Y. Yamamoto, “Bosonic quantum codes for amplitude damping”, Physical Review A 56, 1114 (1997) DOI
- [3]
- M. Ueda, “Probability-density-functional description of quantum photodetection processes”, Quantum Optics: Journal of the European Optical Society Part B 1, 131 (1989) DOI
- [4]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
- [5]
- S. Chessa and V. Giovannetti, “Quantum capacity analysis of multi-level amplitude damping channels”, Communications Physics 4, (2021) arXiv:2008.00477 DOI
- [6]
- M. Grassl et al., “New Constructions of Codes for Asymmetric Channels via Concatenation”, IEEE Transactions on Information Theory 61, 1879 (2015) arXiv:1310.7536 DOI
- [7]
- M. M. Wolf, D. Pérez-García, and G. Giedke, “Quantum Capacities of Bosonic Channels”, Physical Review Letters 98, (2007) arXiv:quant-ph/0606132 DOI
- [8]
- V. Giovannetti and R. Fazio, “Information-capacity description of spin-chain correlations”, Physical Review A 71, (2005) arXiv:quant-ph/0405110 DOI
- [9]
- A. D’Arrigo et al., “Classical and quantum capacities of a fully correlated amplitude damping channel”, Physical Review A 88, (2013) arXiv:1309.2219 DOI
- [10]
- R. Jahangir, N. Arshed, and A. H. Toor, “Quantum capacity of an amplitude-damping channel with memory”, Quantum Information Processing 14, 765 (2014) arXiv:1207.5612 DOI
- [11]
- A. D’Arrigo et al., “Information transmission over an amplitude damping channel with an arbitrary degree of memory”, Physical Review A 92, (2015) arXiv:1510.05313 DOI
- [12]
- R. Lang and P. W. Shor, “Nonadditive quantum error correcting codes adapted to the ampltitude damping channel”, (2007) arXiv:0712.2586
- [13]
- P. W. Shor et al., “High performance single-error-correcting quantum codes for amplitude damping”, (2009) arXiv:0907.5149
- [14]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [15]
- R. Duan et al., “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
- [16]
- T. Jackson, M. Grassl, and B. Zeng, “Concatenated codes for amplitude damping”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) arXiv:1601.07423 DOI
- [17]
- A. S. Fletcher, P. W. Shor, and M. Z. Win, “Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel”, (2007) arXiv:0710.1052
- [18]
- A. Kubica et al., “Erasure Qubits: Overcoming the T1 Limit in Superconducting Circuits”, Physical Review X 13, (2023) arXiv:2208.05461 DOI
- [19]
- Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
- [20]
- J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
- [21]
- K. Sharma et al., “Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels”, New Journal of Physics 20, 063025 (2018) arXiv:1708.07257 DOI
- [22]
- M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018) arXiv:1801.04731 DOI
- [23]
- K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
- [24]
- G. Alber et al., “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
- [25]
- T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI

## 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