Abstract

In classical information theory, polar codes are the first explicit construction of a family of codes that provably achieve the channel capacity for any discrete memoryless classical channel. In this thesis, we investigate the generalizations of polar codes to the case of quantum channels. We start by reviewing the Calderbank-Shor-Steane (CSS) quantum polar coding, which is proposed for quantum systems with dimension 2 (qubits). The CSS quantum polar codes utilize the fact that the recursive construction of polar codes using the quantum CNOT gate yields classical polarization in both the amplitude and phase bases. The first important theorem of this thesis proves a “purely quantum polarization” for quantum channels with dimensions greater than or equal to 2 (qudits). In this polarization, synthesized virtual channels tend to be either completely noiseless or noisy as quantum channels, and not merely in a basis. The channel combining operation for purely quantum polarization is randomly chosen from a finite set of two-qudit unitaries. Taking advantage of this purely quantum polarization phenomenon, we construct an efficient quantum code, where the completely noisy channels are frozen by half of a preshared EPR pair between the encoder and decoder. Hence, our quantum polar code is entanglement assisted. Further, it achieves a quantum communication rate equal to half the symmetric mutual information, which is the symmetric channel capacity for entanglement-assisted quantum communication. Moreover, by chaining several quantum polar codes, we provide a coding scheme, which uses the preshared EPR pairs as catalysts, so that the rate of preshared entanglement vanishes asymptotically.

Subsequently, we focus on an important family of quantum channels known as qubit Pauli channels. Given a Pauli channel, we associate to it a classical channel with a four symbol set as the input alphabet and show that the former polarizes quantumly if and only if the latter polarizes classically. Based on the classical counterpart channel, we provide an alternative proof of quantum polarization for the Pauli channel. More importantly, we also provide an efficient way to decode Pauli errors by decoding the polar code on the classical counterpart channel. We also consider a multilevel polarization for Pauli channels, where polarization happens in multi-levels, such that the synthesized virtual channels can also be “half-noisy” instead of being completely noiseless or noisy. This construction does not use randomization of the channel combining operation. Further, we show that half-noisy channels can be frozen in either the amplitude or the phase basis. Hence, multilevel polarization can be effectively used to construct a quantum polar coding scheme.

Finally, we report on our ongoing work on CSS quantum polar codes in the context of fault-tolerant quantum computing. We provide fault-tolerant procedures for preparing logical encoded quantum states and for error syndrome extraction. Hence, we can protect the encoded logical states for an arbitrarily long time in the low noise limit. Further, for quantum information processing, we provide fault-tolerant procedures to implement the logical Pauli, the Hadamard, and the CNOT gates. Therefore, the only thing missing to have universal fault-tolerant quantum computing with polar codes is a fault-tolerant procedure for the implementation of the T gate.