EAOA qubit stabilizer code

EAOA qubit stabilizer code[1]

Description

Entanglement-assisted qubit stabilizer code in the operator-algebra framework. In the generalized stabilizer formalism of [1], such a code is specified on an extended qubit space by Pauli data consisting of stabilizer, gauge, logical, and sector-labeling operators, and is viewed on the original system as using noiseless ebits shared with a receiver.

EAOA qubit stabilize codes are denoted by \([[n,k; r,e,c_b]]\) or \([[n,k,d; r,e,c_b]]\), where \(n\) is the number of transmitted physical qubits, \(k\) is the number of logical qubits, \(r\) is the number of gauge qubits, \(e\) is the number of ebits, and \(c_b\) is the number of classical strings encoded. When the hybrid component encodes \(c\) classical bits, one has \(c_b=2^c\). This family encompasses ordinary entanglement-assisted qubit stabilizer codes, entanglement-assisted subsystem stabilizer codes, entanglement-assisted hybrid stabilizer codes, and operator-algebra generalizations described within that stabilizer formalism. The framework also exhibits EA hybrid subspace and EA subsystem stabilizer codes that lie outside the earlier EACQ and EAOQECC formalisms [1].

There exist four constructions of EAOA qubit stabilizer codes with distance lower bounds: gauge fixing \([[n,k,d;r,e,c_b]] \to [[n,k,d';r-y,e,\leq 2^y c_b]]\), clean qubits \([[n,k,d;r,0,c_b]] \to [[n-e,k,d';r,e,c_b]]\), entanglement-assisted gauge fixing \([[n,k,d;r,0,c_b]] \to [[n,k,d';r-e,e,c_b]]\), and general gauge fixing \([[n,k,d;r,e,c_b]] \to [[n,k,d';r-y_I-y_S,e+y_S,\leq 2^{y_I}c_b]]\) [1].

Protection

For stabilizer-described qubit subclasses, the EAOAQEC framework yields explicit Pauli error-correction conditions for errors acting on the sender’s qubits under the usual assumption that the receiver’s halves of the ebits are noiseless [1]. Its dressed distance is the minimum weight over logical-centralizer operators outside the isotropic-plus-gauge subgroup together with inter-sector cosets, reducing to the usual distance notions for EAQEC, EAOQECC, EACQ, OAQEC, and related stabilizer code families in the appropriate limits [1; Def. 2 and Rem. 1].

Cousins

Operator-algebra (OA) qubit stabilizer code— EAOA qubit stabilizer codes utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to OA qubit stabilizer codes when said subsystems are interpreted as noiseless physical subsystems.
Hybrid stabilizer code— EA hybrid qubit stabilizer codes utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to hybrid qubit stabilizer codes when said subsystems are interpreted as noiseless physical subsystems. In the original EA hybrid stabilizer formalism, an \([[n,q:c,d;e]]\) EA hybrid stabilizer code is specified by a pair \((\mathcal{S}_Q,\mathcal{S}_C)\) of quantum and classical stabilizer groups, and in the equivalent symplectic formalism by a quantum parity-check matrix together with a classical parity-check matrix [2; Thms. 1-4]. Inside the EAOAQEC stabilizer framework, hybrid stabilizer codes are a proper subclass of the broader EA hybrid subspace codes because the EACQ transversal operators obey additional constraints not required in general [1].
Entanglement-assisted (EA) hybrid QECC— The original EACQ formalism describes a proper subclass of EA hybrid subspace codes inside the EAOAQEC stabilizer framework; EACQ representability imposes extra constraints on the transversal operators beyond belonging to distinct normalizer cosets [1].
Subsystem color code— The 15-qubit subsystem color code yields several EAOA qubit stabilizer constructions, including \([[13,1,3;6,2,3]]\), \([[15,1,3;5,1,2]]\), and \([[15,1,3;4,1,4]]\) examples obtained via clean-qubits and entanglement-assisted gauge-fixing constructions [1].

Member of code lists

Quantum codes

Primary Hierarchy

Quantum Domain

Qubit Kingdom

EAOA qubit codeQuantum

Parents

EAOA qubit code

EAOA qubit stabilizer codes are EAOA qubit codes described within a generalized stabilizer formalism.

EAOA qubit stabilizer code

Children

EA qubit stabilizer code

An EAOA qubit stabilizer code with no gauge or hybrid structure is an EA qubit stabilizer code. Conversely, any \([[n,q+c,d_1;e]]\) EA qubit stabilizer code can be converted into an \([[n,q:c,d_2;e]]\) EA hybrid stabilizer code by repurposing \(c\) logical qubits as classical bits, and any \([[n,q:c,d_2;e]]\) EA hybrid code can be converted into an \([[n,q,d_3;e]]\) EA qubit stabilizer code by absorbing the classical degrees of freedom back into the quantum code, with \(d_1 \leq d_2 \leq d_3\) [2; Thm. 6].

\([[10,1,3;1,3,4]]\) EAOA Hamming code

References

[1]: P. J. Nadkarni, S. Adonsou, G. Dauphinais, D. W. Kribs, and M. Vasmer, “Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction”, (2024) arXiv:2411.14389
[2]: I. Kremsky, M.-H. Hsieh, and T. A. Brun, “Classical enhancement of quantum-error-correcting codes”, Physical Review A 78, (2008) arXiv:0802.2414 DOI

Page edit log

Victor V. Albert (2026-03-27) — most recent

Cite as:

“EAOA qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/eaoa_stabilizer

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