Subsystem CSS code[13] 

Description

Subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Pauli strings. This ensures that the code's stabilizer group is also CSS.

The gauge group generators can be expressed as a matrix using the symplectic reprensetation. This matrix is of the form \begin{align} H=\begin{pmatrix}0 & H_{Z}\\ H_{X} & 0 \end{pmatrix}~. \label{eq:parity} \tag*{(1)}\end{align} The two matrix blocks, \(H_{Z}\) and \(H_X\), correspond to the parity-check matrices of two binary linear codes, an \([n,k_X,d_X]\) code \(C_X\) and \([n,k_Z,d_Z]\) code \(C_Z\), respectively. Code parameters and basis states can be expressed in terms of only data associated with these two classical codes [1,2,4].

Decoding

Steane-type decoder utilizing data from the underlying classical codes [4].

Parents

  • Subsystem qubit stabilizer code — Subsystem CSS codes are subsystem stabilizer codes whose gauge groups admit a generating set of pure-\(X\) and pure-\(Z\) Pauli strings. Additionally, any \([[n,k,r,d]]_{\mathbb{Z}_q}\) subsystem stabilizer code can be mapped onto a \([[2n,2k,2r,\geq d]]_{\mathbb{Z}_q}\) subsystem CSS code, with the mapping preserving geometric locality of a code up to a constant factor [4]. Every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints [4].
  • Subsystem modular-qudit CSS code — Subsystem modular-qudit CSS codes reduce to subsystem qubit CSS codes for \(q=2\).
  • Subsystem Galois-qudit CSS code — Subsystem Galois-qudit CSS codes reduce to subsystem qubit CSS codes for \(q=2\).

Children

Cousin

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

References

[1]
A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
[2]
S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
[3]
S. A. Aly and A. Klappenecker, “Constructions of Subsystem Codes over Finite Fields”, (2008) arXiv:0811.1570
[4]
M. L. Liu, N. Tantivasadakarn, and V. V. Albert, “Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat’s Lemma”, (2023) arXiv:2311.18003
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Zoo Code ID: subsystem_css

Cite as:
“Subsystem CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/subsystem_css
BibTeX:
@incollection{eczoo_subsystem_css, title={Subsystem CSS code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/subsystem_css} }
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Permanent link:
https://errorcorrectionzoo.org/c/subsystem_css

Cite as:

“Subsystem CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/subsystem_css

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