Stabilizer codes : encoding schemes and applications

Abstract

Quantum information science is a rapidly growing research area. It concerns information theory that makes use of quantum nature of the microscopic world. In reality, quantum systems are vulnerable to disturbance from an external environment, which can lead to decoherence in the system. Thus, the system must be protected from the environmental noise to keep information stored in the quantum registers. In order to realize a working quantum computer and dependable quantum information processing, researchers and engineers have to overcome this difficulty. One of the most promising candidates for overcoming decoherence is Quantum Error Correction. The idea of quantum error correction is to protect quantum information from errors due to decoherence and other quantum noise during the transmission of information in quantum channels. One fundamental question of quantum error correction is the existence of quantum error correcting code for a noisy quantum system. Moreover, constructing practical and operational quantum error correcting schemes in actual quantum computing is of great interest to quantum information scientists. In this thesis, stabilizer codes and a scheme for constructing recovery channels without error syndrome detection are studied. The motivation for construction of recovery channel without error syndrome detection is also given. We first review some basic concepts on stabilizer groups and stabilizer codes. In particular, we consider theories and principles involved in the construction of encoding circuits from the generators of stabilizer group, and propose a new procedure to derive recovery channel for a well known quantum code, the [n, k, d] code. First, an algorithm to obtain the generators for a stabilizer code and the corresponding computational basis codewords defined in terms of Pauli operators are reviewed and illustrated in detail. Examples are given to demonstrate the relation between the X- and Z- matrices of generators of stabilizer group and the corresponding encoding circuit. Then based on the general framework of operator quantum error correction, we provide a general scheme on the construction of encoding and decoding circuits for the [n, k, d] codes. Finally, a detailed procedure to construct the recovery channel using encoding circuits and encoded computational basis codewords are demonstrated for [5, 1, 3] code and [8, 3, 3] code step by step as examples, with heuristic explanations based on necessary and sufficient conditions for quantum error correction. Possible future study and open problems will also be mentioned.