Subsystem QECC[1,2] 

Also known as Operator quantum error-correcting code, Gauge quantum error-correcting code.

Description

Quantum code encoding 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} P E^{\dagger}_a E_b P = I_{\mathsf{A}} \otimes g_{ab}^{\mathsf{B}} \tag*{(2)}\end{align} where \(P\) is a projector onto the codespace \(\mathsf{C}\), and \(g_{ab}^{\mathsf{B}}\) is an arbitrary operator on the gauge subsystem.

Notes

See Ref. [4] for an introduction to operator QEC.

Parents

Children

Cousins

  • Quantum error-correcting code (QECC) — A subsystem code reduces to an ordinary error-correcting code when the gauge subsystem is trivial, \(\mathsf{B} = \mathbb{C}\). Conversely, any QECC with a tensor-product logical subspace can be turned into a subsystem code by treating a logical tensor factor as a gauge subsystem.
  • Knill code — Subsystem Knill codes can be formulated [5].
  • Error-transmuting code (QETC) — Subsystem codes are QETCs whose admissible error group decomposes as \(M = I \otimes G\) within the logical and gauge tensor-product space [6; Sec. 4].
  • Bacon-Casaccino subsystem code — Classical information can also be encoded in subsystem codes using their gauge qubits [7].

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 et al., “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]
D. Kribs and D. Poulin, “Operator quantum error correction”, Quantum Error Correction 163 (2013) DOI
[5]
A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
[6]
D. Zhang and T. Cubitt, “Quantum Error Transmutation”, (2023) arXiv:2310.10278
[7]
A. Nemec and A. Klappenecker, “Encoding classical information in gauge subsystems of quantum codes”, International Journal of Quantum Information 20, (2022) DOI
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Zoo Code ID: oecc

Cite as:
“Subsystem QECC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/oecc
BibTeX:
@incollection{eczoo_oecc, title={Subsystem QECC}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/oecc} }
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“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.