Quantum Polarization of Qudit Channels

Abstract

We provide a generalization of quantum polar codes to quantum channels with qudit-input, achieving the symmetric coherent information of the channel. Our scheme relies on a channel combining and splitting construction, where a two-qudit unitary randomly chosen from a unitary 2-design is used to combine two instances of a qudit-input channel. The inputs to the synthesized bad channels are frozen by sharing EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that the generalized two-qudit Clifford group forms a unitary 2-design, therefore the channel combining operation can be chosen from this set. Moreover, we show that polarization also happens for a much smaller subset of two-qudit Cliffords, which is not a unitary 2-design. Finally, we show how to decode the proposed quantum polar codes on Pauli qudit channels.

Quantum polar stabilizer codes based on polarization of pure quantum channel are bad stabilizer codes for quantum computing

“…In this paper, we try to design a class of QECCs for quantum computing, whose coding rate asymptotically achieve the quantum channel capacity. Some researchers [30,31] believe that directly quantizing classical polar coding circuits will produce polarization phenomenon of pure quantum channel, based on which we can design a quantum polar coding scheme that applies to quantum computing. Following this intuition, we proved polarization of two-dimensional-input quantum symmetric channels (QSCs) in our previous work [32], according to which we propose a class of quantum polar stabilizer codes (QPSCs).…”

Channel Polarization of Two-dimensional-input Quantum Symmetric Channels

“…I of [6] means a classical binary quasi symmetric channel), the symmetric capacity is its Shannon capacity. However, up to the present, none of the previous studies [31][32][33][34][35][36][37][38][39][40][41] has proved that the symmetric coherent information of a pure quantum channel is its MSLCI. Next, we will prove this theorem for twodimensional-input QQSC.…”

“…Some previous studies [31][32][33][34][35][36][38][39][40][41] have proved some quantities of classical-quantum channels whose inputs are classical bits and outputs are qubits, such as classical symmetric capacity [31], Bhattacharyya parameter [31,41], and classical symmetric Holevo information [39] will polarize. Some studies [32,33,35,40,41] has referred to coherent information, which is a quantum quantity of quantum channels and is believed to be a quantity to measure the channel capacity of pure quantum channels [42][43][44][45][46][47][48][49][50][51], but the coherent information of the classical-quantum channels is just the classical mutual information. Based on the polarization of classicalquantum channels, researchers have proposed some quantum polar coding schemes.…”