Welcome to the Lattice Kingdom.

Lattice-based code Encodes states (codewords) in coordinates of a lattice in the \(n\)-dimensional real coordinate space \(\mathbb{R}^n\). The number of codewords may be infinite because the coordinate space is infinite-dimensional, so various restricted versions have to be constructed in practice. Since lattices are closed under addition, lattice-based codes can be thought of as linear codes over the reals. Parents: Error-correcting code (ECC). Parent of: Niset-Andersen-Cerf code. Cousins: Linear code. Cousin of: Group-based code, Multi-mode GKP code.
Niset-Andersen-Cerf code[1] Quantum-inspired classical code encoding two-mode coherent states \(\{|\alpha\rangle, |\beta\rangle\}\) into four modes such that the complex values \((\alpha,\beta)\) are recoverable after a single-mode erasure. There are two variations of the storage procedure: a deterministic protocol that offers recovery against a single mode erasure, and a probabalistic that can protect against multiple errors with post selection. This code is effectively protecting classical information stored in \((\alpha,\beta)\) using quantum operations. Protection: The deterministic protocol protects against a single erasure error on a known mode. This recovers one state perfectly and the other state with fidelity \(F = \frac{1}{1 + e^{-2 r}}\) for an initial EPR pair squeezed with variance \(e^{-2r}\). The probabalistic protocol utilizes post-selection to protect against multiple erasures with state-dependent fidelity. Parents: Lattice-based code. Cousin of: Homological bosonic code.

References

[1]
J. Niset, U. L. Andersen, and N. J. Cerf, “Experimentally Feasible Quantum Erasure-Correcting Code for Continuous Variables”, Physical Review Letters 101, (2008). DOI; 0710.4858