# Compass code[1]

## Description

Subspace or subsystem CSS code defined by gauge-fixing the Bacon-Shor code, i.e., the code whose gauge group consists of terms in the compass model Hamiltonian [2–4] on a square lattice. Families of random codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise.

The gauge fixing proceeds by denoting plaquettes by \(X\) or \(Z\) type using two colors, and fixing or cutting the corresponding \(X\) or \(Z\)-type gauge generators at the respective plaquettes. A fully colored lattice yields a subspace code, but allowing for non-colored plaquettes yield a subsystem code. A fully non-colored lattice reduces to the Bacon-Shor code.

The surface-density compass code family is obtained by randomly cutting \(X\)-type stabilizers at only plaquettes of one color in a checkboard coloring; it interpolates between Bacon-Shor codes and rotated surface codes. The Shor-density compass code family is obtained by randomly cutting \(X\)-type stabiilzers at any plaquette; it interpolates between Bacon-Shor codes and QPCs.

## Protection

## Decoding

## Code Capacity Threshold

## Parent

## Children

- Bacon-Shor code — A compass code on a fully non-colored lattice reduces to the Bacon-Shor code.
- Heavy-hexagon code — The heavy-hex code can be viewed as a compass code if ancilla qubits are ignored [6].

## Cousins

- Rotated surface code — The surface-density compass code family interpolates between Bacon-Shor codes and rotated surface codes.
- Quantum parity code (QPC) — The Shor-density compass code family interpolates between Bacon-Shor codes and QPCs.
- Random stabilizer code — Compass code families are constructed by randomly assigning stabilizers to plaquettes of a square lattice.
- Asymmetric quantum code — Families of random compass codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise [1].

## References

- [1]
- M. Li et al., “2D Compass Codes”, Physical Review X 9, (2019) arXiv:1809.01193 DOI
- [2]
- K. I. Kugel’ and D. I. Khomskiĭ, “The Jahn-Teller effect and magnetism: transition metal compounds”, Soviet Physics Uspekhi 25, 231 (1982) DOI
- [3]
- J. Dorier, F. Becca, and F. Mila, “Quantum compass model on the square lattice”, Physical Review B 72, (2005) arXiv:cond-mat/0501708 DOI
- [4]
- Z. Nussinov and J. van den Brink, “Compass and Kitaev models -- Theory and Physical Motivations”, (2013) arXiv:1303.5922
- [5]
- B. Pato, J. W. Staples Jr., and K. R. Brown, “Logical coherence in 2D compass codes”, (2024) arXiv:2405.09287
- [6]
- C. Chamberland et al., “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020) arXiv:1907.09528 DOI

## Page edit log

- Victor V. Albert (2024-03-01) — most recent

## Cite as:

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