Subsystem modular-qudit stabilizer code 

Also known as Gauge modular-qudit stabilizer code.

Description

Modular-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a modular qudit stabilizer code and assigning some of its logical qubits to be gauge qubits. For composite qudit dimensions, such codes need not encode an integer number of qudits.

Subsystem stabilizer codes are defined by a gauge group \(\mathsf{G}\) and a stabilizer group \(\mathsf{S}\), both subgroups of the \(n\)-qudit Pauli group \(\mathsf{P}_n\) that satisfy \(\mathsf{Z}(\mathsf{G})=\mathsf{S}\), where \(\mathsf{Z}\) denotes taking the center of a group.

A code can be constructed by starting with either group. Given an \(\mathsf{S}\), one can pick any \(\mathsf{G}\) satisfying \(\mathsf{S}\subseteq\mathsf{G}\subseteq\mathsf{N(S)}\), where \(\mathsf{N(S)}\) is the normalizer of the stabilizer group within \(\mathsf{P}_n\). Alternatively, given a \(\mathsf{G}\), one defines \(\mathsf{S}\) to be the center of the gauge group.

The logical Pauli group is \(\mathsf{N(G)/S}\). As such, the case when \(\mathsf{G}=\mathsf{S}\) reduces to an ordinary stabilizer code, while the case \(\mathsf{G}=\mathsf{N(S)}\) reduces to a trivial code.

One can gauge fix [1] an Abelian subgroup of the gauge group by adding it to the stabilizer group.

Gauge fixing: Gauge fixing is a map between subsystem codes that is done using an Abelian subgroup \(\mathsf{F}\subseteq\mathsf{G}\), \begin{align} \begin{split} \mathsf{S}&\to\left\langle \mathsf{S},\mathsf{F}\right\rangle \\ \mathsf{G}&\to\mathsf{N}_{\mathsf{G}}\left(\mathsf{F}\right)~, \end{split} \tag*{(1)}\end{align} where \(\mathsf{N}_{\mathsf{G}}\left(\mathsf{F}\right)\) is the normalizer of the stabilizer group within \(\mathsf{G}\).

Gauge fixing can be used to switch between different stabilizer codes that yield different gauge sets in a process known as gauge switching. Gauge fixing also encompasses lattice surgery and code deformation [2].

One can also gauge out a subgroup \(\mathsf{F}\) of the Pauli group by adding it to the gauge group.

Gauging out: Gauging out is a map between subsystem codes that is done using a subgroup \(\mathsf{F}\subseteq\mathsf{P}_n\), \begin{align} \begin{split} \mathsf{S}&\to\mathsf{Z}\left(\left\langle \mathsf{G},\mathsf{F}\right\rangle \right)\\ \mathsf{G}&\to\left\langle \mathsf{G},\mathsf{F}\right\rangle ~. \end{split} \tag*{(2)}\end{align} The stabilizer group of the output subsystem code is a subgroup of that of the input code, \(\mathsf{Z}\left(\left\langle \mathsf{G},\mathsf{F}\right\rangle \right)\subseteq\mathsf{Z}\left(\mathsf{G}\right)\). When \(\mathsf{F}\) is a subgroup of the logical Pauli group, this is also called gauging. If \(\mathsf{F}\) is itself a Pauli group of \(m\) logical qubits of the original subsystem code, then gauging those qubits is equivalent to treating them as gauge qubits.

Decoding

Syndrome measurements are obtained by first measuring gauge operators of the code and taking their products, which give the stabilizer measurement outcomes. The order in which gauge operators are measured is important since they do not commute. There is a sufficient condition for inferring the stabilizer syndrome from the measurements of the gauge generators [3; Appendix].Decoder for certain geometrically local subsystem codes from hypergraphs [4].

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  • Modular-qudit stabilizer code — Subsystem modular-qudit stabilizer codes reduce to modular-qudit stabilizer codes when there are no gauge qudits.

References

[1]
D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005) arXiv:quant-ph/0508131 DOI
[2]
C. Vuillot et al., “Code deformation and lattice surgery are gauge fixing”, New Journal of Physics 21, 033028 (2019) arXiv:1810.10037 DOI
[3]
M. Suchara, S. Bravyi, and B. Terhal, “Constructions and noise threshold of topological subsystem codes”, Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011) arXiv:1012.0425 DOI
[4]
V. V. Gayatri and P. K. Sarvepalli, “Decoding Algorithms for Hypergraph Subsystem Codes and Generalized Subsystem Surface Codes”, (2018) arXiv:1805.12542
[5]
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: qudit_subsystem_stabilizer

Cite as:
“Subsystem modular-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_subsystem_stabilizer
BibTeX:
@incollection{eczoo_qudit_subsystem_stabilizer, title={Subsystem modular-qudit stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_subsystem_stabilizer} }
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“Subsystem modular-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_subsystem_stabilizer

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