## 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 also 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]. For unital noise channels, such codes are related to the multiplicative domain of the channel [6].

## Encoding

## Realizations

## Notes

## Parent

- Operator-algebra QECC (OAQECC) — An OAQECC which has gauge structure (e.g., gauge qubits) but no block structure is a subsystem QECC.

## Children

## Cousins

- Quantum error-correcting code (QECC) — A subsystem QECC reduces to an ordinary (i.e., subspace) QECC when the gauge subsystem is trivial. 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) subsystem QECC — EA subsystem 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.
- Quantum-double code — Subsystem versions of quantum-double codes have been formulated [10].
- Knill code — Subsystem Knill codes can be formulated [11].
- Quantum 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 [12; Sec. 4].
- Subsystem hypergraph product (SHP) code — Classical information can also be encoded in subsystem codes using their gauge qubits [13].

## 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]
- 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]
- 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
- [7]
- O. Oreshkov, “Robustness of operator quantum error correction with respect to initialization errors”, Physical Review A 77, (2008) arXiv:0709.3533 DOI
- [8]
- K. M. Schreiter et al., “Optical implementation of a unitarily correctable code”, Physical Review A 80, (2009) arXiv:0909.1584 DOI
- [9]
- D. Kribs and D. Poulin, “Operator quantum error correction”, Quantum Error Correction 163 (2013) DOI
- [10]
- P. Kumar, “A Class of Quantum Double Subsystem Codes”, (2011) DOI
- [11]
- A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
- [12]
- D. Zhang and T. Cubitt, “Quantum Error Transmutation”, (2023) arXiv:2310.10278
- [13]
- 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

- Victor V. Albert (2022-03-02) — most recent
- Srilekha Gandhari (2021-12-14)
- Victor V. Albert (2021-11-24)

## 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.