Operator-algebra (OA) qubit code 

Also known as Hybrid subsystem 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, reduces to a hybrid code when \(\mathsf{B}_{1,2}\) are both trivial, and reduces to an ordinary (i.e., subspace) qubit code when \(\mathsf{B_1}\) and \(\mathsf{A}_{2}\otimes\mathsf{B}_{2}\) 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.