Towards efficient decoding of classical-quantum polar codes

TL;DR

This paper proposes a more practical decoding strategy for classical-quantum polar codes that reduces the need for complex collective measurements by combining non-collective processing with selective collective measurements, achieving capacity.

Contribution

It introduces a decoding approach that recovers most bits with non-collective measurements and employs collective measurements only for the remaining bits, improving practicality.

Findings

Non-collective decoding recovers N I(W_acc) bits

Collective measurements are necessary for the remaining bits

Explicit Helstrom measurement forms for small polar codes

Abstract

Known strategies for sending bits at the capacity rate over a general channel with classical input and quantum output (a cq channel) require the decoder to implement impractically complicated collective measurements. Here, we show that a fully collective strategy is not necessary in order to recover all of the information bits. In fact, when coding for a large number N uses of a cq channel W, N I(W_acc) of the bits can be recovered by a non-collective strategy which amounts to coherent quantum processing of the results of product measurements, where I(W_acc) is the accessible information of the channel W. In order to decode the other N (I(W) - I(W_acc)) bits, where I(W) is the Holevo rate, our conclusion is that the receiver should employ collective measurements. We also present two other results: 1) collective Fuchs-Caves measurements (quantum likelihood ratio measurements) can be used at the receiver to achieve the Holevo rate and 2) we give an explicit form of the Helstrom measurements used in small-size polar codes. The main approach used to demonstrate these results is a quantum extension of Arikan's polar codes.