# Quantum polar code[1]

## Description

Entanglement-assisted CSS code utilized in a quantum polar coding scheme producing entangled pairs of qubits between sender and receiver. In such a scheme, the amplitude and phase information of a quantum state is handled in complementary fashion [2] using an encoding based on classical polar codes. Variants of the initial scheme have been developed for degradable channels [3] and extended to arbitrary channels [4].

The scheme requires some a-priori quantum side information in the general case, making the associated code entanglement assisted [1]. The requirement of having quantum side information vanishes when the sum of the amplitude channel fidelity and the phase channel fidelity is not greater than 1. It is shown to vanish for the case of degradable noise channels [4]. A more complicated quantum polar-coding scheme that does not require pre-shared entanglement has also been derived [5].

## Protection

## Rate

## Decoding

## Fault Tolerance

## Parent

- EA qubit stabilizer code — Quantum polar codes are CSS codes used in an entanglement generation scheme that generally requires entanglement assistance. They require assistance only to determine positions to store information which optimally protect against both bit and phase noise. Without this assistance, they are just CSS codes constructed out of polar codes. A variant of quantum polar codes exists that does not require entanglement assistance [5].

## Cousins

- Polar code — Without entanglement assistance, quantum polar codes are just CSS codes constructed out of polar codes.
- Coherent-state constellation code — Coherent-state constellation codes consisting of points from a Gaussian quadrature rule can be concatenated with quantum polar codes to achieve the Gaussian coherent information of the thermal noise channel [7][8].
- \([[4,2,2]]\) CSS code — \([[4,2,2]]\) code is a small quantum polar code [9].

## References

- [1]
- J. M. Renes, F. Dupuis, and R. Renner, “Efficient Polar Coding of Quantum Information”, Physical Review Letters 109, (2012) arXiv:1109.3195 DOI
- [2]
- J. M. Renes and J.-C. Boileau, “Physical underpinnings of privacy”, Physical Review A 78, (2008) arXiv:0803.3096 DOI
- [3]
- M. M. Wilde and J. M. Renes, “Quantum polar codes for arbitrary channels”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1201.2906 DOI
- [4]
- M. M. Wilde and S. Guha, “Polar Codes for Degradable Quantum Channels”, IEEE Transactions on Information Theory 59, 4718 (2013) arXiv:1109.5346 DOI
- [5]
- J. M. Renes et al., “Efficient Quantum Polar Codes Requiring No Preshared Entanglement”, IEEE Transactions on Information Theory 61, 6395 (2015) arXiv:1307.1136 DOI
- [6]
- A. Goswami, M. Mhalla, and V. Savin, “Fault-Tolerant Preparation of Quantum Polar Codes Encoding One Logical Qubit”, (2023) arXiv:2209.06673
- [7]
- F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent-state constellations and polar codes for thermal Gaussian channels”, Physical Review A 95, (2017) arXiv:1603.05970 DOI
- [8]
- F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent state constellations for Bosonic Gaussian channels”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) DOI
- [9]
- Kyungjoo Noh, Leung code as quantum polar code, 2017.

## Page edit log

- Victor V. Albert (2022-07-06) — most recent
- Richard Barney (2022-05-18)
- Victor V. Albert (2021-12-03)

## Cite as:

“Quantum polar code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_polar

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/quantum_polar.yml.