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Subsystem qubit code

Alternative Names: Gauge qubit code.
Root code for the Qubit Kingdom

Description

Subsystem QECC encoding into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space.

Transversal and Permutation-Based Gates

If a subsystem qubit code \(Q\) of length \(n\) has compact subgroups \(N\triangleleft G\leq \mathrm{Aut}(Q)\) such that \(G/N\) is finite, non-Abelian, simple, and not \(A_5\), then \(n\) is at least the minimal permutation degree \(\mu(G/N)\) [1; Thm. 1].

Cousin

  • Qubit code— Subsystem qubit codes reduce to (subspace) qubit codes when there is no gauge subsystem.

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Parents
OA qubit codeQuantum
An OA qubit code which has gauge structure (e.g., gauge qubits) but no block structure is a subsystem qubit code.
Subsystem modular-qudit codeSubsystem QECC Quantum
Subsystem modular-qudit codes reduce to subsystem qubit codes for qudit dimension \(q=2\).
Subsystem Galois-qudit codeSubsystem QECC Quantum
Subsystem Galois-qudit quantum codes for \(q=2\) correspond to subsystem qubit codes.
Subsystem qubit code
Children
Subsystem qubit stabilizer code

References

[1]
A. S. Morris and D. Malz, “Constraints on phantom codes from automorphism group bounds”, (2026) arXiv:2604.15111
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Zoo Code ID: subsystem_qubits_into_qubits

Cite as:
“Subsystem qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/subsystem_qubits_into_qubits, arXiv:2606.11484
BibTeX:
@incollection{eczoo_subsystem_qubits_into_qubits,
title={Subsystem qubit code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/subsystem_qubits_into_qubits}
}
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Permanent link:
https://errorcorrectionzoo.org/c/subsystem_qubits_into_qubits

Cite as:

“Subsystem qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/subsystem_qubits_into_qubits, arXiv:2606.11484

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