# Lattice subsystem code[1]

Also known as Topological subsystem code.

## Description

A geometrically local qubit, modular-qudit, or Galois-qudit subsystem stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\). On an infinite lattice, its gauge and stabilizer groups are generated by few-site Pauli operators and their translations, in which case the code is called translationally invariant subsystem code. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions, in which case the stabilizer group may no longer be generated by few-site Pauli operators. Lattice defects and boundaries between different codes can also be introduced.

## Protection

The subsystem BT bound is an upper bound of \(d = O(L^{D-1})\) on the distance [2] of lattice subsystem stabilizer codes arranged in a \(D\)-dimensional lattice of length \(L\) with \(n=L^D\). In particular, \(D=2\)-dimensional subsystem codes satisfy \(kd = O(n)\) [3]. More generally, there is a tradeoff theorem [4] stating that, for any logical operator, there is an equivalent logical operator with weight \(\tilde{d}\) such that \(\tilde{d}d^{1/(D-1)}=O(L^{D})\).

## Parent

## Children

- Doubled color code
- Subsystem color code
- Generalized five-squares code
- Heavy-hexagon code
- Three-fermion (3F) subsystem code
- 3D subsystem surface code
- Subsystem rotated surface code
- Subsystem surface code
- \(\mathbb{Z}_q^{(1)}\) subsystem code
- Chiral semion subsystem code
- \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code

## Cousins

- Lattice stabilizer code — Lattice subsystem codes reduce to lattice stabilizer codes when there are no gauge qudits. The former (latter) is required to admit few-site gauge-group (stabilizer-group) generators on a lattice with boundary conditions.
- Abelian topological code — All 2D Abelian bosonic topological orders can be realized as modular-qudit lattice subsystem codes by starting with an Abelian quantum double model (slightly different from that of Ref. [5]) along with a family of Abelian TQDs that generalize the double semion anyon theory and gauging out certain bosonic anyons [6]. The stabilizer generators of the new subsystem code may no longer be geometrically local. Non-Abelian topological orders are purported not to be realizable with Pauli stabilizer codes [7].

## References

- [1]
- H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
- [2]
- S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes”, New Journal of Physics 11, 043029 (2009) arXiv:0810.1983 DOI
- [3]
- S. Bravyi, “Subsystem codes with spatially local generators”, Physical Review A 83, (2011) arXiv:1008.1029 DOI
- [4]
- J. Haah and J. Preskill, “Logical-operator tradeoff for local quantum codes”, Physical Review A 86, (2012) arXiv:1011.3529 DOI
- [5]
- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
- [6]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [7]
- A. C. Potter and R. Vasseur, “Symmetry constraints on many-body localization”, Physical Review B 94, (2016) arXiv:1605.03601 DOI

## Page edit log

- Victor V. Albert (2024-01-30) — most recent

## Cite as:

“Lattice subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/translationally_invariant_subsystem