Description
A code whose logical subspace is the joint eigenspace (usually with eigenvalue \(+1\)) of a set of commuting unitary Pauli-type operators forming the code's stabilizer group. They can be block codes defined of tensor-product spaces of qubits or qudits, or non-block codes defined on single sufficiently large Hilbert spaces such as bosonic modes or group spaces.
The coding theory motivation for stabilizer codes came from linear binary codes, whose codewords form a closed subspace in the space of binary strings. Stabilizer codes extend this property, in various ways, to quantum error correction. Stabilizer codes can be defined succinctly using the stabilizer group generators and without explicitly writing out a basis of codewords. The stabilizer formalism is applicable to almost all quantum-code kingdoms; see list of stabilizer codes for a list of all stabilizer codes.
Stabilizer codes were originally defined for qubits, where the relevant commuting operators are tensor products of Pauli matrices. The Pauli stabilizer structure is useful in providing standardized encoding, gates, decoding, and performance bounds. Elements of this structure remain in qudit extensions, in particular for prime-dimensional modular qudits and Galois qudits. Infinite-dimensional Pauli-type bases yield the bosonic stabilizer and rotor stabilizer codes.
Extensions of the stabilizer formalism, such as XS and XP stabilizer codes, relax the mutual commutation property. Other extensions, such as CWS and union stabilizer codes, enlarge the codespace by re-assigning error words as codewords.
Protection
The group of all Pauli-type operators typically serves as the set of noise operators for stabilizer codes.Gates
A Gottesman-Knill-type theorem exists for qubits, modular qudits, Galois qudits, and rotors [1,2], as well as oscillators [3–5].Decoding
The structure of stabilizer codes allows for straightforward syndrome-based decoding because the stabilizer generators serve as the code's check operators, and their eigenvalues serve as the error syndromes. The error correction process involves measuring the stabilizer generators and applying correcting Pauli-type operators based on the measurement outcomes.Cousins
- Abelian topological code— There is a general correspondence between stabilizer codes and gauge theory, with the stabilizer group playing the role of the gauge group [6].
- Majorana stabilizer code— Majorana stabilizer codes are useful for Majorana-based architectures, where the degrees of freedom are electrons, and the notion of locality is different than all other code kingdoms.
Member of code lists
Primary Hierarchy
References
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- [2]
- J. Bermejo-Vega, C. Y.-Y. Lin, and M. V. den Nest, “The computational power of normalizer circuits over black-box groups”, (2014) arXiv:1409.4800
- [3]
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- [5]
- A. Mari and J. Eisert, “Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient”, Physical Review Letters 109, (2012) arXiv:1208.3660 DOI
- [6]
- S. Carrozza, A. Chatwin-Davies, P. A. Hoehn, and F. M. Mele, “A correspondence between quantum error correcting codes and quantum reference frames”, (2024) arXiv:2412.15317
- [7]
- Yu. I. Manin, “Some remarks on Koszul algebras and quantum groups”, Annales de l’institut Fourier 37, 191 (1987) DOI
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- Y. I. Manin, “Quantized Theta-Functions”, Progress of Theoretical Physics Supplement 102, 219 (2013) DOI
- [9]
- Yu. I. Manin, “Functional equations for quantum theta functions”, (2003) arXiv:math/0307393
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- E. Chang-Young and H. Kim, “Theta vectors and quantum theta functions”, Journal of Physics A: Mathematical and General 38, 4255 (2005) arXiv:math/0402401 DOI
- [11]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [12]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
- [13]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, 45 arXiv:quant-ph/0703112 DOI
Page edit log
- Victor V. Albert (2022-04-19) — most recent
Cite as:
“Stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stabilizer