Description
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}~. \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.
Parents
Children
- Entanglement-assisted (EA) operator QECC — EQ operator QECCs utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to subsystem QECCs when said subsystems are interpreted as noiseless physical subsystems.
- Subsystem modular-qudit stabilizer code
Cousin
- 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.
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
Page edit log
- Victor V. Albert (2022-03-02) — most recent
- Srilekha Gandhari (2021-12-14)
- Victor V. Albert (2021-11-24)
Cite as:
“Subsystem quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/oecc
Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/oecc.yml.