# Polarization of Quantum Channels using Clifford-based Channel Combining

## TL;DR

This paper introduces a quantum polar coding scheme using Clifford gates, achieving optimal information transmission for qubit channels with entanglement assistance and efficient decoding.

## Contribution

It presents a novel quantum polar code construction utilizing Clifford-based channel combining, achieving symmetric coherent information and entanglement-assisted communication.

## Key findings

- Achieves symmetric coherent information for qubit channels.
- Uses a small set of nine Clifford gates for polarization.
- Provides an efficient decoding algorithm for Pauli channels.

## Abstract

We provide a purely quantum version of polar codes, achieving the symmetric coherent information of any qubit-input quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from a much smaller subset of only nine two-qubit Clifford gates. Subsequently, we show that a Pauli channel polarizes if and only if a specific classical channel over a four-symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels. Finally, we present a code construction based on chaining several quantum polar codes, which is shown to require a rate of preshared entanglement that vanishes asymptotically.