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].

Cousin

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

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Subsystem qubit codeSubsystem QECC Quantum

Subsystem qubit stabilizer codeSubsystem QECC Quantum

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 codeSubsystem QECC Quantum

Subsystem modular-qudit CSS codes reduce to subsystem qubit CSS codes for \(q=2\).

Subsystem Galois-qudit CSS codeSubsystem QECC Quantum

Subsystem Galois-qudit CSS codes reduce to subsystem qubit CSS codes for \(q=2\).

Subsystem CSS code

Children

Bravyi-Bacon-Shor (BBS) code

Compass code

Subsystem lifted-product (SLP) code

Subsystem homological product code

Subsystem color code

CSS-Plaquette code

3D subsystem surface code

Subsystem hyperbolic surface code

Subsystem rotated surface code

Subsystem surface code

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.