Amplitude-damping (AD) code

Amplitude-damping (AD) code[1,2]

Description

Block quantum code on either qubits or bosonic modes that is designed to detect and correct qubit or bosonic AD errors, respectively.

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. The fixed point of this channel for any truncation of Fock space is unique [5].

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 [6].

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 [7].

Rate

The quantum capacity of the AD channel is \(\max\{0, \log \frac{1-\gamma}{\gamma}\} \) [8]. Quantum capacities of the qubit AD channel are also determined [9,10], including of channels with memory [11,12]. Capacities of qudit extensions have also been studied [6].

Cousins

Qubit CSS code— An \([[n,k,d_Z=t+1,d_X=2t+1]]\) qubit CSS code protects against \(t\) AD errors [14][13; Sec. 8.7].
Concatenated qubit code— Concatenated quantum codes can protect against qubit AD [15].
Dual-rail quantum code— Dual-rail qubits can be used to convert leakage and AD noise into erasure noise [16,17]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [18] that protects against \(d-1\) AD errors [14].
Gottesman-Kitaev-Preskill (GKP) code— Particular families of GKP codes achieve the capacity of AD and amplification channels [19].
\([[2m,2m-2,2]]\) error-detecting code— The \([[2m,2m-2,2]]\) error-detecting code [20] and its relative the code with single stabilizer \(XX\cdots X\) [21] admit continuous-time QEC against single AD errors.
\([[4,2,2]]\) Four-qubit code— The \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto 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 [18,22,23]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [14].
Qubit stabilizer code— Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [18] that protects against \(d-1\) AD errors [14].

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Approximate operator-algebra QECC

Operator-algebra QECC (OAQECC)

Quantum error-correcting code (QECC)

Block quantum code

Parents

Block quantum code

Approximate quantum error-correcting code (AQECC)Quantum

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].

Amplitude-damping (AD) code

Children

Fock-state bosonic code

Fock-state codes are designed to protect against bosonic AD noise.

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.

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 qubit code

Self-complementary quantum codes consisting of computational basis states whose bitstrings are sufficiently spaced apart correct at least one AD error [24; Thm. 2.5][25; Thm. 2].

Post-selected PI code

Qudit GNU PI code

Qudit GNU PI codes protect against AD errors.

Numerically optimized four-qubit AD code

References

[1]: D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, “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. G. Schirmer and X. Wang, “Stabilizing open quantum systems by Markovian reservoir engineering”, Physical Review A 81, (2010) arXiv:0909.1596 DOI
[6]: S. Chessa and V. Giovannetti, “Quantum capacity analysis of multi-level amplitude damping channels”, Communications Physics 4, (2021) arXiv:2008.00477 DOI
[7]: M. Grassl, P. W. Shor, G. Smith, J. Smolin, and B. Zeng, “New Constructions of Codes for Asymmetric Channels via Concatenation”, IEEE Transactions on Information Theory 61, 1879 (2015) arXiv:1310.7536 DOI
[8]: M. M. Wolf, D. Pérez-García, and G. Giedke, “Quantum Capacities of Bosonic Channels”, Physical Review Letters 98, (2007) arXiv:quant-ph/0606132 DOI
[9]: V. Giovannetti and R. Fazio, “Information-capacity description of spin-chain correlations”, Physical Review A 71, (2005) arXiv:quant-ph/0405110 DOI
[10]: A. D’Arrigo, G. Benenti, G. Falci, and C. Macchiavello, “Classical and quantum capacities of a fully correlated amplitude damping channel”, Physical Review A 88, (2013) arXiv:1309.2219 DOI
[11]: 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
[12]: A. D’Arrigo, G. Benenti, G. Falci, and C. Macchiavello, “Information transmission over an amplitude damping channel with an arbitrary degree of memory”, Physical Review A 92, (2015) arXiv:1510.05313 DOI
[13]: D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[14]: R. Duan, M. Grassl, Z. Ji, and B. Zeng, “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
[15]: 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
[16]: A. S. Fletcher, P. W. Shor, and M. Z. Win, “Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel”, (2007) arXiv:0710.1052
[17]: A. Kubica, A. Haim, Y. Vaknin, H. Levine, F. Brandão, and A. Retzker, “Erasure Qubits: Overcoming the T1 Limit in Superconducting Circuits”, Physical Review X 13, (2023) arXiv:2208.05461 DOI
[18]: Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
[19]: G. Zheng, W. He, G. Lee, K. Noh, and L. Jiang, “Performance and achievable rates of the Gottesman-Kitaev-Preskill code for pure-loss and amplification channels”, (2024) arXiv:2412.06715
[20]: C. Ahn, H. Wiseman, and K. Jacobs, “Quantum error correction for continuously detected errors with any number of error channels per qubit”, Physical Review A 70, (2004) arXiv:quant-ph/0402067 DOI
[21]: C. Ahn, H. M. Wiseman, and G. J. Milburn, “Quantum error correction for continuously detected errors”, Physical Review A 67, (2003) arXiv:quant-ph/0302006 DOI
[22]: G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger, “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
[23]: T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
[24]: R. Lang and P. W. Shor, “Nonadditive quantum error correcting codes adapted to the ampltitude damping channel”, (2007) arXiv:0712.2586
[25]: P. W. Shor, G. Smith, J. A. Smolin, and B. Zeng, “High performance single-error-correcting quantum codes for amplitude damping”, (2009) arXiv:0907.5149

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/ampdamp.yml.