Operator-algebra (OA) qubit stabilizer code[1]
Description
An OAQECC 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 \(j,k\), \begin{align} E_{j}^{\dagger}E_{k}&\notin(\mathsf{N(S)}-\mathsf{G})\cup\tag*{(1)}\\& \,\,\,\,\,\,\,\,\,\,\,\,\,\cup\left(\bigcup_{\tau,\sigma:\tau\mathsf{N(S)}\neq\sigma\mathsf{N(S)}}\tau\mathsf{N(S)}\sigma\right)~. \tag*{(2)}\end{align}
Parent
Children
- Hybrid stabilizer code — An OA stabilizer code which has no gauge qubits but has a block structure that corresponds to a linear binary code is a hybrid stabilizer code.
- Qubit stabilizer code — An OA qubit stabilizer code storing no classical information and admitting no gauge qubits is a qubit stabilizer code.
- Subsystem qubit stabilizer code — An OA qubit stabilizer code storing no classical information but retaining gauge qubits for its quantum code is a subsystem qubit stabilizer code.
Cousin
- Bacon-Shor code — The OA qubit stabilizer formalism yields hybrid Bacon-Shor codes [1].
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
Page edit log
- Michael Liu (2023-17-06) — most recent
Cite as:
“Operator-algebra (OA) qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), NaN. https://errorcorrectionzoo.org/c/qubit_stabilizer_oaqecc