Multilevel polarization for quantum Channels

Recently, a purely quantum version of polar codes has been proposed in~\cite{DGMS19} based on channel combining and splitting procedure, where randomly chosen two-qubit Clifford unitary acts as operation. Here, we consider the code construction using same procedure in~\cite{DGMS19}, but with fixed unitary. For family Pauli channels, show that polarization happens in multi-levels, synthesized virtual channels tend to become completely noisy, half-noisy, or noiseless. Further, present exploiting multilevel nature polarization, provide an efficient decoding for this code. We half-noisy can be frozen by fixing their inputs either amplitude phase basis, which allows reducing number preshared EPR pairs compared in~\cite{DGMS19}. upper bound pairs, is equality case erasure channel. To improve speed propose alternative construction, again polarizes previous also holds. channel, confirm numerical analysis relatively faster construction.