  The successive cancellation list decoder (SCL) is an efficient decoder for
classical polar codes with low decoding error, approximating the maximum
likelihood decoder (MLD) for small list sizes. Here we adapt the SCL to the
task of decoding quantum polar codes and show that it inherits the high
performance and low complexity of the classical case, and can approximate the
quantum MLD for certain channels. We apply SCL decoding to a novel version of
quantum polar codes based on the polarization weight (PW) method, which
entirely avoids the need for small amounts of entanglement assistance apparent
in previous quantum polar code constructions. When used to find the precise
error pattern, the quantum SCL decoder (SCL-E) shows competitive performance
with surface codes of similar size and low-density parity check codes of
similar size and rate. The SCL decoder may instead be used to approximate the
probability of each equivalence class of errors, and then choose the most
likely class. We benchmark this class-oriented decoder (SCL-C) against the
SCL-E decoder and find a noticeable improvement in the logical error rate. This
improvement stems from the fact that the contributions from just the low-weight
errors give a reasonable approximation to the error class probabilities. Both
SCL-E and SCL-C maintain the complexity O(LN logN) of SCL for code size N and
list size L. We also show that the list decoder can be used to gain insight
into the weight distribution of the codes and how this impacts the effect of
degenerate errors.
