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 \(n\)-modular-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 modular-qudit 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 qudits of the original subsystem code, then gauging out those qudits is equivalent to treating them as gauge qubits. Gauging out should not be confused with gauging (or ungauging) symmetries [3–6], a different process rooted in gauge theory which can be done to stabilizer or subsystem codes and which can change \(n\).
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 [7; Appendix].Decoder for certain geometrically local subsystem codes from hypergraphs [8].Cousin
- Modular-qudit stabilizer code— Subsystem modular-qudit stabilizer codes reduce to modular-qudit stabilizer codes when there are no gauge qudits.
Member of code lists
Primary Hierarchy
References
- [1]
- D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005) arXiv:quant-ph/0508131 DOI
- [2]
- C. Vuillot, L. Lao, B. Criger, C. García Almudéver, K. Bertels, and B. M. Terhal, “Code deformation and lattice surgery are gauge fixing”, New Journal of Physics 21, 033028 (2019) arXiv:1810.10037 DOI
- [3]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [4]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [5]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [6]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [7]
- 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
- [8]
- V. V. Gayatri and P. K. Sarvepalli, “Decoding Algorithms for Hypergraph Subsystem Codes and Generalized Subsystem Surface Codes”, (2018) arXiv:1805.12542
- [9]
- M. L. Liu, N. Tantivasadakarn, and V. V. Albert, “Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma”, Quantum 8, 1403 (2024) arXiv:2311.18003 DOI
Page edit log
- Victor V. Albert (2022-11-09) — most recent
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