Hayden-Nezami-Salton-Sanders bosonic code[1]
Description
An \([[n,1]]_{\mathbb{R}}\) analog CSS code defined using homological structures 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-trivial code with 5 modes [1].Cousins
- Generalized homological-product CSS code— Hayden-Nezami-Salton-Sanders 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)— Hayden-Nezami-Salton-Sanders codes have been considered in the context of spacetime replication of quantum data [1,2], while STCs are designed to replicate classical data.
Primary Hierarchy
References
- [1]
- P. Hayden, S. Nezami, G. Salton, and B. C. Sanders, “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:
“Hayden-Nezami-Salton-Sanders bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hnss