  Polar coding, introduced 2008 by Arikan, is the first (very) efficiently
encodable and decodable coding scheme whose information transmission rate
provably achieves the Shannon bound for classical discrete memoryless channels
in the asymptotic limit of large block sizes. Here we study the use of polar
codes for the transmission of quantum information. Focusing on the case of
qubit Pauli channels and qubit erasure channels, we use classical polar codes
to construct a coding scheme which, using some pre-shared entanglement,
asymptotically achieves a net transmission rate equal to the coherent
information using efficient encoding and decoding operations and code
construction. Furthermore, for channels with sufficiently low noise level, we
demonstrate that the rate of preshared entanglement required is zero.
