Subsystem quantum error-correcting code[1][2]


A subsystem code, also known as an operator QECC or gauge QECC, 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}~. \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}} \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.




  • 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.
  • Entanglement-assisted (EA) operator QECC — Entanglement-assisted operator QECCs are subsystem QECCs utilizing pre-shared entanglement.

Zoo code information

Internal code ID: oecc

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Zoo Code ID: oecc

Cite as:
“Subsystem quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_oecc, title={Subsystem quantum error-correcting code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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D. Kribs, R. Laflamme, and D. Poulin, “Unified and Generalized Approach to Quantum Error Correction”, Physical Review Letters 94, (2005). DOI; quant-ph/0412076
David W. Kribs et al., “Operator quantum error correction”. quant-ph/0504189
M. A. Nielsen and D. Poulin, “Algebraic and information-theoretic conditions for operator quantum error correction”, Physical Review A 75, (2007). DOI; quant-ph/0506069

Cite as:

“Subsystem quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.