  We present a scheme for magic state distillation using punctured polar codes.
Our results build on some recent work by Bardet et al. (ISIT, 2016) who
discovered that polar codes can be described algebraically as decreasing
monomial codes. Using this powerful framework, we construct tri-orthogonal
quantum codes (Bravyi et al., PRA, 2012) that can be used to distill magic
states for the $T$ gate. An advantage of these codes is that they permit the
use of the successive cancellation decoder whose time complexity scales as
$O(N\log(N))$. We supplement this with numerical simulations for the erasure
channel and dephasing channel. We obtain estimates for the dimensions and error
rates for the resulting codes for block sizes up to $2^{20}$ for the erasure
channel and $2^{16}$ for the dephasing channel. The dimension of the
triply-even codes we obtain is shown to scale like $O(N^{0.8})$ for the binary
erasure channel at noise rate $0.01$ and $O(N^{0.84})$ for the dephasing
channel at noise rate $0.001$. The corresponding bit error rates drop to
roughly $8\times10^{-28}$ for the erasure channel and $7 \times 10^{-15}$ for
the dephasing channel respectively.
