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, the encoded quantum information in subsystem \(\mathsf{A}\) can be recovered modulo an arbitrary error on the auxiliary or gauge subsystem \(\mathsf{B}\). The gauge subsystem provides additional freedom to the error correction process: errors that act trivially on the information subsystem \(\mathsf{A}\) but nontrivially on \(\mathsf{B}\) need not be corrected. The subsystem \(\mathsf{B}\) can encode gauge qubits when its dimension is a power of two. While all operator QECCs are also ordinary QECCs, the attachment of a gauge subsystem to a code allows for a wider variety of encoding procedures, fault-tolerant logical operations, and efficient error-correction protocols.
Protection
The necessary and sufficient error-correction conditions are, for all errors \(E_a,E_b\) in an error set \(\cal{E}\) [3]: \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 \(\mathsf{B}\). This condition ensures that distinguishing and correcting errors based on their effect on subsystem \(\mathsf{A}\) is sufficient; errors that act identically on \(\mathsf{A}\) but differ on \(\mathsf{B}\) are considered equivalent.
These can be studied in the presence of continuous noise [4].
A unitarily recoverable subsystem is a correctable subsystem whose logical information can be restored by a unitary operation, possibly into a different subsystem representation; thus, recovery is more relaxed than correction [5]. In fact, every correctable subsystem is unitarily recoverable [5; Thm. 1]. For unital noise channels, unitarily correctable subsystems are precisely the noiseless subsystems of \(\mathcal{E}^{\dagger}\circ\mathcal{E}\) [5; Thm. 2]; these are related to the multiplicative domain of the channel [6] (see also [7]).
No additional OQEC conditions are needed to tolerate imperfect initialization: under the standard subsystem-code conditions, the effective noise induced by population outside the code can only increase the fidelity with an ideally encoded state, and this robustness persists under encoded CPTP operations [8; Thms. 3,4].
Encoding
Subsystem QECCs are robust to initialization errors without modifying the standard OQEC conditions [8].Decoding
Petz recovery map provides a recovery operation that is near-optimal for certain subsystem codes [9].Realizations
A two-qubit unitarily recoverable subsystem code recovery has been realized in an optical system [10].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.
- Knill code— Subsystem Knill codes can be formulated [14].
- 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 [15; Sec. 4].
Primary Hierarchy
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, R. Laflamme, D. Poulin, and M. Lesosky, “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]
- A. Peres, “Unitary dynamics for quantum codewords”, (1996) arXiv:quant-ph/9609015
- [8]
- O. Oreshkov, “Robustness of operator quantum error correction with respect to initialization errors”, Physical Review A 77, (2008) arXiv:0709.3533 DOI
- [9]
- P. Mandayam and H. K. Ng, “Towards a unified framework for approximate quantum error correction”, Physical Review A 86, (2012) arXiv:1202.5139 DOI
- [10]
- K. M. Schreiter, A. Pasieka, R. Kaltenbaek, K. J. Resch, and D. W. Kribs, “Optical implementation of a unitarily correctable code”, Physical Review A 80, (2009) arXiv:0909.1584 DOI
- [11]
- D. W. Kribs, “A quantum computing primer for operator theorists”, (2004) arXiv:math/0404553
- [12]
- D. W. Kribs, “A brief introduction to operator quantum error correction”, (2005) arXiv:math/0506491
- [13]
- D. Kribs and D. Poulin, “Operator quantum error correction”, Quantum Error Correction 163 (2013) DOI
- [14]
- A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
- [15]
- D. Zhang and T. Cubitt, “Quantum Error Transmutation”, (2023) arXiv:2310.10278
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.