# Homological bosonic code[1]

## Description

An \([[n,1]]_{\mathbb{R}}\) analog CSS code defined using homological structres associated with an \(n-1\) simplex. Relevant to the study of spacetime replication of quantum information [2].

Stabilizer generators are defined by two orthogonal subspaces of the \(C_1\) in the chain complex. \(C_X = \partial_2 C_2\) and \(C_P = \partial_1^T Q\) for some \(Q \subset C_0\). The standard approach would use \(Q = C_0\), which would mean the logical dimension would be the dimension of the 1st homology group \(H^1\). However, \(H^1\) is trivial for the \(n-1\) simplex, so one chooses \(Q \neq C_0\) such that exactly one stabilizer is removed, yielding a stabilizer code instead of a single stabilized state.

## Protection

## Encoding

## Decoding

## Notes

## Parent

## Cousins

- Generalized homological-product CSS code — Homological CV codes utilize chain complexes in code construction, but the complexes have trivial homology.
- Niset-Andersen-Cerf code — The Niset-Andersen-Cerf code can be viewed as a scheme to replicate quantum information in multiple regions [1].
- Spacetime code (STC) — Homological CV codes have been considered in the context of spacetime replication of quantum data [1,2], while STCs are designed to replicate classical data.

## References

- [1]
- P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016) arXiv:1601.02544 DOI
- [2]
- P. Hayden and A. May, “Summoning information in spacetime, or where and when can a qubit be?”, Journal of Physics A: Mathematical and Theoretical 49, 175304 (2016) arXiv:1210.0913 DOI

## Page edit log

- Victor V. Albert (2022-04-21) — most recent
- Victor V. Albert (2022-01-04)
- Siddharth Taneja (2021-12-19)

## Cite as:

“Homological bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/homological_cv