Multilevel Polarization for Quantum Channels

TL;DR

This paper introduces a multilevel quantum polarization technique for quantum channels, demonstrating improved code construction and reduced entanglement resources, with faster polarization in certain channels.

Contribution

It presents a fixed-unitary quantum polar code with multilevel polarization, an efficient decoding method, and an improved construction that speeds up polarization.

Findings

Polarization occurs in multiple levels for Pauli channels.

Half-noisy channels can be frozen to reduce EPR pairs.

The alternative construction speeds up polarization in quantum erasure channels.

Abstract

Recently, a purely quantum version of polar codes has been proposed in [1] based on a quantum channel combining and splitting procedure, where a randomly chosen two-qubit Clifford unitary acts as channel combining operation. Here, we consider the quantum polar code construction using the same channel combining and splitting procedure as in [1], but with a fixed two-qubit Clifford unitary. For the family of Pauli channels, we show that polarization happens in multi-levels, where synthesized quantum virtual channels tend to become completely noisy, half-noisy, or noiseless. Further, we present a quantum polar code exploiting the multilevel nature of polarization, and provide an efficient decoding for this code. We show that half-noisy channels can be frozen by fixing their inputs in either the amplitude or the phase basis, which allows reducing the number of preshared EPR pairs compared to the construction in [1]. We provide an upper bound on the number of preshared EPR pairs, which is an equality in the case of the quantum erasure channel. To improve the speed of polarization, we propose an alternative construction, which again polarizes in multi-levels, and the previous upper bound on the number of preshared EPR pairs also holds. For a quantum erasure channel, we confirm by numerical analysis that the multilevel polarization happens relatively faster for the alternative construction.