Subsystem QECC[1,2] 

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


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.


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.


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




  • 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].


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
D. W. Kribs et al., “Operator quantum error correction”, (2006) arXiv:quant-ph/0504189
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
D. Kribs and D. Poulin, “Operator quantum error correction”, Quantum Error Correction 163 (2013) DOI
A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
D. Zhang and T. Cubitt, “Quantum Error Transmutation”, (2023) arXiv:2310.10278
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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Cite as:
“Subsystem QECC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_oecc, title={Subsystem QECC}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Subsystem QECC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.