# Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs

## TL;DR

This paper extends polarization theory to arbitrary classical-quantum channels and cq-MACs, enabling efficient polar code construction with provably fast error decay.

## Contribution

It proves polarization for arbitrary cq-channels using Abelian groups and constructs polar codes for these channels and cq-MACs with efficient encoding and decoding.

## Key findings

- Polarization to deterministic homomorphism channels is achieved.
- Encoder complexity is O(N log N).
- Error probability decays faster than 2^{-N^{β}} for any β<1/2.

## Abstract

We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation and an Ar{\i}kan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels which project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple access channels (cq-MAC). The encoder can be implemented in $O(N\log N)$ operations, where $N$ is the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than $2^{-N^{\beta}}$ for any $\beta<\frac{1}{2}$.