# Homological bosonic code[1]

## Description

An \([[n,1]]_{\mathbb{R}}\) oscillator-into-oscillator CSS stabilizer 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

## Realizations

## Parents

- Oscillator-into-oscillator code
- Bosonic stabilizer code — Homological CV codes are bosonic CSS codes.

## Cousins

- Calderbank-Shor-Steane (CSS) stabilizer code — CSS and homological CV codes utilize chain complexes in code construction, with the latter complexes having 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 [2][1], while STCs are designed to replicate classical data.

## Zoo code information

## References

- [1]
- P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016). DOI; 1601.02544
- [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). DOI; 1210.0913

## Cite as:

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