Quantum polar stabilizer codes based on polarization of pure quantum channel don't work for quantum computing

TL;DR

This paper investigates quantum polar stabilizer codes inspired by classical polar codes, finds that their current construction methods are ineffective for quantum computing, and proposes a future direction for designing high-rate quantum stabilizer codes.

Contribution

It introduces a new quantum polar stabilizer code construction algorithm and demonstrates its ineffectiveness through simulations, providing insights for future code design.

Findings

Quantum polar stabilizer codes from the proposed algorithm do not perform effectively.

The second intuition for quantum polar codes leads to a dead end.

A new class of quantum stabilizer codes with 0.5 coding rate is identified for certain noise models.

Abstract

Inspired by classical polar codes, whose coding rate can asymptotically achieve the Shannon capacity, researchers are trying to find its analogue in quantum information field, which are called quantum polar codes. However, no one has designed a quantum polar coding scheme which applies to quantum computing yet. There are two intuitions in previous research. The first is that directly converting classical polar coding circuits to quantum ones will produce polarization phenomenon of pure quantum channel, which has been proved in our previous work. The second is that based on this quantum polarization phenomenon one can design a quantum polar coding scheme that applies to quantum computing. There are several previous work following the second intuition, none of which has been verified by experiments. In this paper, we follow the second intuition and propose a more reasonable quantum polar stabilizer code construction algorithm than any previous ones by using the theory of stabilizer codes. Unfortunately, simulation experiments show that even the stabilizer codes obtained from this more reasonable construction algorithm don't work, which implies that the second intuition leads to a dead end. Based on the analysis on why the second intuition don't work, we provide a possible future direction of designing quantum stabilizer codes with high coding rate by borrowing the idea of classical polar codes. following this direction, we find a class of quantum stabilizer codes with coding rate 0.5 for pure Pauli X, Z and Y noise.