  We provide a purely quantum version of polar codes, achieving the symmetric
coherent information of any qubit-input quantum channel. Our scheme relies on a
recursive channel combining and splitting construction, where a two-qubit gate
randomly chosen from the Clifford group is used to combine two single-qubit
channels. The inputs to the synthesized bad channels are frozen by preshared
EPR pairs between the sender and the receiver, so our scheme is entanglement
assisted. We further show that quantum polarization can be achieved by choosing
the channel combining Clifford operator randomly, from a much smaller subset of
only nine two-qubit Clifford gates. Subsequently, we show that a Pauli channel
polarizes if and only if a specific classical channel over a four-symbol input
set polarizes. We exploit this equivalence to prove fast polarization for Pauli
channels, and to devise an efficient successive cancellation based decoding
algorithm for such channels. Finally, we present a code construction based on
chaining several quantum polar codes, which is shown to require a rate of
preshared entanglement that vanishes asymptotically.
