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 \(C_1\) 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 cohomology 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 arbitrarily 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