The Coherent Parity Check Framework for Quantum Error Correction

Abstract: Quantum error correction protocols are an essential element in the design of any circuit-model quantum computer. In this thesis, I introduce the coherent parity check (CPC) framework for quantum error correction. CPC codes have a fundamental structure in which quantum parity check measurements are stored coherently and compared over time. The specific advantage of the CPC code structure is that it provides a way of creating new stabilizer codes from the starting point of any sequence of parity checks. I show that this freedom in the choice of parity checks can be used to derive methods for the construction of distance-three quantum codes based on almost any distance-three classical code. The CPC framework has further applications in machine search routines for code discovery, as well as in the design of bespoke codes tailored for the demands of a given device. Another feature of CPC codes is that they can be represented as factor graphs of the type commonly seen in classical error correction and machine learning. I outline a procedure for this mapping, and demonstrate how a quantum code can be derived by manipulating its factor graph representation. The aim of the factor graph mapping for CPC codes is to make it easier to adapt well-developed techniques from classical information theory for use with quantum codes. This will make the CPC framework a useful tool for the theoretical and practical study of quantum error correction codes as large-scale quantum computers move closer to becoming a reality.