# Lattice subsystem code[1]

## Description

## Rate

Subsystem BT bound: 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 \(\tilde{d}d^{1/(D-1)}=O(L^{D})\).

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## Gates

Subsystem PYBK bound: The Bravyi-Koenig bound can be extended to subsystem codes by Pastawski and Yoshida. Namely, logical gates implemented via constant-depth quantum circuits on a \(D\)-dimensional lattice subsystem code whose distance increases at least logarithmically with \(n\) lie in the \(D\)th level of the Clifford hierarchy [5].

## Parent

- Sparse subsystem code — Lattice subsystem codes are sparse subsystem codes on Euclidean geometries.

## Children

- Generalized five-squares code
- 2D subsystem color code
- 3D subsystem color code
- Doubled color code
- Heavy-hexagon code
- Three-fermion (3F) subsystem code
- 3D subsystem surface code
- Subsystem rotated surface code
- Subsystem surface code
- Modular-qudit subsystem color code — Modular-qudit subsystem color codes are defined analogous to qubit subsystem color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizers commute [6; Sec. VII].
- \(\mathbb{Z}_q^{(1)}\) 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. [7]) along with a family of Abelian TQDs that generalize the double semion anyon theory and gauging out certain bosonic anyons [8]. 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 [9].
- Majorana subsystem stabilizer code — Translationally invariant subsystem codes have been formulated in terms of Majorana operators [10].

## 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]
- F. Pastawski and B. Yoshida, “Fault-tolerant logical gates in quantum error-correcting codes”, Physical Review A 91, (2015) arXiv:1408.1720 DOI
- [6]
- F. H. E. Watson et al., “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
- [7]
- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
- [8]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [9]
- A. C. Potter and R. Vasseur, “Symmetry constraints on many-body localization”, Physical Review B 94, (2016) arXiv:1605.03601 DOI
- [10]
- A. Chapman, S. T. Flammia, and A. J. Kollár, “Free-Fermion Subsystem Codes”, (2022) arXiv:2201.07254

## 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