Subsystem CSS code[13] 


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].


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


  • 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 \(q=2\).



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


A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
S. A. Aly and A. Klappenecker, “Constructions of Subsystem Codes over Finite Fields”, (2008) arXiv:0811.1570
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.
@incollection{eczoo_subsystem_css, title={Subsystem CSS code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

“Subsystem CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.