Multilevel Polarization for Quantum Channels

Recently, a purely quantum version of polar codes has been proposed in [3] based on a quantum channel combining and splitting procedure, where a randomly chosen two-qubit Clifford unitary acts as channel combining operation. Here, we consider the quantum polar code construction using the same channel combining and splitting procedure as in [3] but with a fixed two-qubit Clifford unitary. For the family of Pauli channels, we show that the polarization happens although in multilevels, where synthesised quantum virtual channels tend to become completely noisy, half-noisy or noiseless. Further, it is shown that half-noisy channels can be frozen by fixing their inputs in either amplitude or phase basis, which reduces the number of preshared EPR pairs with respect to the construction in [3]. We also give an upper bound on the number of preshared EPR pairs, which is an equality in the case of quantum erasure channel. To improve the speed of polarization, we provide an alternative construction, which again polarizes in multilevel way and the earlier upper bound on preshared EPR pairs also holds. We confirm by numerical analysis for a quantum erasure channel that the multilevel polarization happens relatively faster for the alternative construction.