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Subsystem QECC[1,2]

Alternative names: Operator QECC (OQECC), Gauge QECC.

Description

A quantum code which encodes quantum information in a tensor factor of a subspace that is decomposed into a tensor product of subsystems.

A subsystem code encodes information in a subsystem \(\mathsf{A}\) of the code space \(\mathsf{C}\), which is part of the system Hilbert space \(\mathsf{H}\), as \begin{align} \mathsf{H}=\mathsf{C} \oplus \mathsf{C}^{\perp} = \mathsf{A} \otimes \mathsf{B} \oplus \mathsf{C}^{\perp}~. \tag*{(1)}\end{align} Following an error, it is sufficient to revert back to the original state modulo a transformation on the auxiliary or gauge subsystem \(\mathsf{B}\). The subsystem \(\mathsf{B}\) therefore gives additional freedom to the error correction process, and is said to encode gauge qubits when its dimension is a power of two. While strictly speaking all operator QECCs are also ordinary QECCs, the attachment of a subsystem to a code allows for a wider variety of encoding procedures, fault-tolerant logical operations, and efficient error-correction protocols.

Protection

Necessary and sufficient [3] error-correction conditions are, for all errors \(E_a,E_b\) in an error set \(\cal{E}\), \begin{align} \Pi E^{\dagger}_a E_b \Pi = I_{\mathsf{A}} \otimes g_{ab}^{\mathsf{B}} \tag*{(2)}\end{align} where \(\Pi\) is a projector onto the codespace \(\mathsf{C}\), and \(g_{ab}^{\mathsf{B}}\) is an arbitrary operator on the gauge subsystem. These have been studied in the presence of continuous noise [4].

A unitarily correctable subsystem is a subsystem code whose encoded information can be recovered via a unitary, i.e., in a measurement-free way [5] (see also [6]). For unital noise channels, such codes are related to the multiplicative domain of the channel [7].

Encoding

Subsystem QECCs are robust to initialization errors [8].

Decoding

Petz recovery map is shown to be near-optimal for certain subsystem codes [9].

Realizations

A two-qubit unitarily correctable subsystem code recovery has been realized in an optical system [10].

Notes

See Refs. [1113] for an introduction to operator QEC.

Cousins

Primary Hierarchy

References

[1]
D. Kribs, R. Laflamme, and D. Poulin, “Unified and Generalized Approach to Quantum Error Correction”, Physical Review Letters 94, (2005) arXiv:quant-ph/0412076 DOI
[2]
D. W. Kribs, R. Laflamme, D. Poulin, and M. Lesosky, “Operator quantum error correction”, (2006) arXiv:quant-ph/0504189
[3]
M. A. Nielsen and D. Poulin, “Algebraic and information-theoretic conditions for operator quantum error correction”, Physical Review A 75, (2007) arXiv:quant-ph/0506069 DOI
[4]
O. Oreshkov, D. A. Lidar, and T. A. Brun, “Operator quantum error correction for continuous dynamics”, Physical Review A 78, (2008) arXiv:0806.3145 DOI
[5]
D. W. Kribs and R. W. Spekkens, “Quantum error-correcting subsystems are unitarily recoverable subsystems”, Physical Review A 74, (2006) arXiv:quant-ph/0608045 DOI
[6]
A. Peres, “Unitary dynamics for quantum codewords”, (1996) arXiv:quant-ph/9609015
[7]
M.-D. Choi, N. Johnston, and D. W. Kribs, “The multiplicative domain in quantum error correction”, Journal of Physics A: Mathematical and Theoretical 42, 245303 (2009) arXiv:0811.0947 DOI
[8]
O. Oreshkov, “Robustness of operator quantum error correction with respect to initialization errors”, Physical Review A 77, (2008) arXiv:0709.3533 DOI
[9]
P. Mandayam and H. K. Ng, “Towards a unified framework for approximate quantum error correction”, Physical Review A 86, (2012) arXiv:1202.5139 DOI
[10]
K. M. Schreiter, A. Pasieka, R. Kaltenbaek, K. J. Resch, and D. W. Kribs, “Optical implementation of a unitarily correctable code”, Physical Review A 80, (2009) arXiv:0909.1584 DOI
[11]
D. W. Kribs, “A quantum computing primer for operator theorists”, (2004) arXiv:math/0404553
[12]
D. W. Kribs, “A brief introduction to operator quantum error correction”, (2005) arXiv:math/0506491
[13]
D. Kribs and D. Poulin, “Operator quantum error correction”, Quantum Error Correction 163 (2013) DOI
[14]
P. Kumar, “A Class of Quantum Double Subsystem Codes”, (2011) DOI
[15]
A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
[16]
D. Zhang and T. Cubitt, “Quantum Error Transmutation”, (2023) arXiv:2310.10278
[17]
A. Nemec and A. Klappenecker, “Encoding classical information in gauge subsystems of quantum codes”, International Journal of Quantum Information 20, (2022) DOI
Page edit log

Cite as:

“Subsystem QECC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/oecc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oecc.yml.