Subsystem CSS code

Subsystem CSS code[1–3]

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} G=\begin{pmatrix}0 & G_{Z}\\ G_{X} & 0 \end{pmatrix}~. \label{eq:parity} \tag*{(1)}\end{align} The two matrix blocks, \(G_{Z}\) and \(G_X\), 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].

Protection

For any code whose gauge group is generated by \(XX\) and \(ZZ\), the weight of an \(X\)-type (\(Z\)-type) single-qubit bare-logical operator is lower-bounded by the number of \(Z\)-type (\(X\)-type) bare-logical operators acting on its supporting logical qubits [5,6].

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. Any \([[n,k,r,d]]\) subsystem stabilizer code can be mapped onto a \([[2n,2k,2r,\geq d]]\) subsystem CSS code via symplectic doubling, which preserves geometric locality of a code up to a constant factor. 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&apos;s Lemma”, Quantum 8, 1403 (2024) arXiv:2311.18003 DOI
[5]: M. Marvian and S. Lloyd, “Robust universal Hamiltonian quantum computing using two-body interactions”, (2019) arXiv:1911.01354
[6]: P. Lisonek, A. Roy, and S. Trandafir, private communication, 2019

Page edit log

Victor V. Albert (2023-11-13) — most recent

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.