Operator-algebra (OA) qubit code 

Root code for the Qubit Kingdom

Description

An OAQECC family that encompasses ordinary (i.e., subspace) qubit codes, subsystem qubit codes, and hybrid qubit codes using a unified operator-algebraic framework.

A simple example encompassing elements of all three subfamilies encodes a single logical qubit and a single classical bit into a direct sum of two subsystem qubit codes. A quantum subsystem code \(\mathsf{A}_j\otimes\mathsf{B}_j\), with \(\mathsf{A}_j\) the logical qubit factor, and \(\mathsf{B}_j\) the gauge qubit factor, is associated with each of the two values \(j\in\{0,1\}\) of the classical bit. The corresponding decomposition of the Hilbert space \(\mathsf{H}\) is \begin{align} \mathsf{H}=(\mathsf{A}_{1}\otimes\mathsf{B}_{1})\oplus(\mathsf{A}_{2}\otimes\mathsf{B}_{2})\oplus\mathsf{C}^{\perp}~, \tag*{(1)}\end{align} where \(\mathsf{C}^\perp\) is the combined error space of both codes. The above code reduces to a subsystem code when \(\mathsf{A}_{2}\otimes\mathsf{B}_{2}\) is trivial, and reduces to a hybrid code when \(\mathsf{B}_{1,2}\) are both trivial.

Parent

Children

  • Hybrid qubit code — An OA qubit code which has no which has no gauge structure (e.g., gauge qubits) but has a block structure that corresponds to a classical code is a hybrid qubit code.
  • Operator-algebra (OA) qubit stabilizer code
  • Qubit code — An OA qubit code which has no gauge qubits and no block structure is a qubit code.
  • Subsystem qubit code — An OA qubit code which has gauge structure (e.g., gauge qubits) but no block structure is a subsystem qubit code.
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Zoo Code ID: oa_qubits_into_qubits

Cite as:
“Operator-algebra (OA) qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/oa_qubits_into_qubits
BibTeX:
@incollection{eczoo_oa_qubits_into_qubits, title={Operator-algebra (OA) qubit code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/oa_qubits_into_qubits} }
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Permanent link:
https://errorcorrectionzoo.org/c/oa_qubits_into_qubits

Cite as:

“Operator-algebra (OA) qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/oa_qubits_into_qubits

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