Operator-algebra qubit stabilizer code[1] 

Root code for the Qubit Kingdom

Description

Operator-algebra QECC in which the commutant \(\mathcal{A}'\) of the logical algebra \(\mathcal{A}\) arises as the group algebra of a subgroup \(\mathsf{G}\) of the \(n\)-qubit Pauli group \(\mathsf{P}_n\). The stabilizer \(\mathsf{S}\) is the center of \(\mathsf{G}\) modulo factors of \(i I\). The quotient \(\mathsf{P}_n / \mathsf{N(S)}\), where \(\mathsf{N(S)}\) is the normalizer of \(\mathsf{S}\), is in bijective correspondence with the factors of the logical algebra \(\mathcal{A}\).

Protection

Specialized conditions for the correctability of \(\mathcal{A}\) with respect to an error operation \(\mathcal{E}\) with operation elements \(\{E_j\}_j\) can be given in group theoretic terms. Indeed, \(\mathcal{A}\) is correctable for \(\mathcal{E}\) if, for all \(j,k\), \[E_j^\dagger E_k \notin (\mathsf{N(S)} - \mathsf{G}) \bigcup \big(\bigcup_{\tau, \sigma: \tau \mathsf{N(S)} \neq \sigma \mathsf{N(S)}} \tau \mathsf{N(S)} \sigma\big)~.\]

Parent

Children

References

[1]
G. Dauphinais, D. W. Kribs, and M. Vasmer, “Stabilizer Formalism for Operator Algebra Quantum Error Correction”, Quantum 8, 1261 (2024) arXiv:2304.11442 DOI
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Zoo Code ID: qubit_stabilizer_oaqecc

Cite as:
“Operator-algebra qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), NaN. https://errorcorrectionzoo.org/c/qubit_stabilizer_oaqecc
BibTeX:
@incollection{eczoo_qubit_stabilizer_oaqecc, title={Operator-algebra qubit stabilizer code}, booktitle={The Error Correction Zoo}, year={NaN}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qubit_stabilizer_oaqecc} }
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Cite as:

“Operator-algebra qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), NaN. https://errorcorrectionzoo.org/c/qubit_stabilizer_oaqecc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/subsystem/qubit_stabilizer_oaqecc.yml.