# Niset-Andersen-Cerf code[1]

## Description

Coherent-state c-q 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.

## Encoding

After an EPR pair preparation, use 2 continuous CNOT and 2 continuous inverse CNOT gates to entangle a bosonic EPR pair with initial states \(|\alpha \rangle\) and \(|\beta \rangle\).Alternate optical encoder using a two-mode squeezed vacuum state and two balanced beam splitters to mix the input coherent states with the EPR pair.

## Decoding

Optical decoder using three beam splitters, electronic gain detectors, and two phase-insensitive amplifiers as described in Ref. [1].

## Realizations

Realized in Ref. [2] in an optical system with 3 beam-splitters. The fidelity peaked around \(0.6\) for deterministic approach, and around \(0.77\) for the probabilistic approach (with a 25% chance of error).

## Parent

## Cousins

- Quadrature-amplitude modulation (QAM) code — The Niset-Andersen-Cerf code encodes two coherent states at a time with arbitrary complex values, making it analogous to a two-point QAM code. The code does not encode any quantum information since superpositions of the coherent states are not stored. However, analysis of the code is done via a quantum treatment.
- Homological bosonic code — The Niset-Andersen-Cerf code can be viewed as a scheme to replicate quantum information in multiple regions [3].

## 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) arXiv:0710.4858 DOI
- [2]
- M. Lassen et al., “Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code”, Nature Photonics 4, 700 (2010) arXiv:1006.3941 DOI
- [3]
- P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016) arXiv:1601.02544 DOI

## Page edit log

- Victor V. Albert (2022-11-08) — most recent
- Victor V. Albert (2022-07-06)
- Victor V. Albert (2022-01-04)
- Siddharth Taneja (2021-12-19)

## Cite as:

“Niset-Andersen-Cerf code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/niset_andersen_cerf