Homological bosonic code

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

Protects against certain types of erasure errors (depending on the specific dimension). Certain constructions also protect arbitrary sized errors on multiple photon states.

Encoding

Encoding depends on the specific dimension, but can generally be done using generalized conditional-rotation and Fourier-transform gates.

Decoding

Decoding requires a different circuit for each possible erasure error, with no general circuit decoding any possible erasure error. Every circuit relies on a generalized conditional rotation, which Ref. [1] calls the QND Gate and which is defined as \(QND_c | x , y \rangle = |x + c y, y \rangle\).

Notes

Proposed experimental optical procedure for realizing the simplest non-trival code with 5 modes [1].

Parent

Analog stabilizer code

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/stabilizer/hyperplane/homological_cv.yml.