Subsystem modular-qudit CSS code 

Description

Modular-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) modular-qudit 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 \(q\)-ary linear codes, an \([n,k_X,d_X]_q\) code \(C_X\) and \([n,k_Z,d_Z]_q\) code \(C_Z\), respectively. For prime-dimensional qudits, code parameters and code basis states have been expressed in terms of only data associated with these two classical codes [1,2].

Symplectic doubling: 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 [2] (see also [3][4; Thm. 1]). In the symplectic representation, the gauge-group generator matrix of the former is mapped into that of latter as follows, \begin{align} \begin{pmatrix}G_{X} & G_{Z}\end{pmatrix} \to \begin{pmatrix} 0 & 0 & G_{Z} & -G_{X}\\ G_{X} & G_{Z} & 0 & 0 \end{pmatrix}~, \tag*{(2)}\end{align} where the first two columns of the latter matrix correspond to the \(X\)-type part of the gauge-group generator matrix of the output subsystem CSS code. In the case of a stabilizer code, the stabilizer generator matrix is mapped instead (see [4; Thm. 1] for the case of qubit stabilizer codes). For geometrically local 2D subsystem codes, this mapping yields a twisted double cover of the underlying qudit geometry [5].

Decoding

Steane-type decoder utilizing data from the underlying classical codes [2].

Parent

  • Subsystem modular-qudit stabilizer code — Subsystem modular-qudit CSS codes are subsystem modular-qudit stabilizer codes whose gauge groups admit a generating set of pure-\(X\) and pure-\(Z\) Pauli strings. 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 via symplectic doubling, which preserves geometric locality of a code up to a constant factor. Every subsystem prime-qudit stabilizer code can be constructed from two nested subsystem prime-qudit CSS codes satisfying certain constraints [2].

Children

Cousin

  • Modular-qudit CSS code — Subsystem modular-qudit CSS codes reduce to (subspace) modular-qudit CSS codes when there is no gauge subsystem.

References

[1]
S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
[2]
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
[3]
S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
[4]
A. A. Kovalev and L. P. Pryadko, “Quantum Kronecker sum-product low-density parity-check codes with finite rate”, Physical Review A 88, (2013) arXiv:1212.6703 DOI
[5]
S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: qudit_subsystem_css

Cite as:
“Subsystem modular-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qudit_subsystem_css
BibTeX:
@incollection{eczoo_qudit_subsystem_css, title={Subsystem modular-qudit CSS code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_subsystem_css} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/qudit_subsystem_css

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/subsystem/qudit_subsystem_css.yml.