Fault-tolerant syndrome extraction techniques for quantum error correction

Quantum computing promises revolutionary advances in computational power for problems such as factorization, unstructured search, and quantum simulation. However, quantum systems are inherently fragile and subject to decoherence and operational errors. Quantum error correction (QEC) schemes and fault-tolerant quantum error correction (FTQEC) are essential to mitigate these issues and enable reliable quantum information processing. This thesis focuses on fault-tolerant syndrome extraction, an integral part of QEC, exploring various methods that aim to detect and correct errors without disturbing the encoded quantum information. Special attention is given to flag-based and cat-state-based syndrome extraction schemes, which provide efficient fault-tolerant while reducing resource overhead. Theoretical foundations are established through a review of stabilizer codes and CSS codes included color code and surface code, followed by an analysis of teoretical implementations using gadgets such as Steane, Knill, and flag circuits. The work further investigates advanced decoding algorithms, such as minimum weight perfect matching(MWPM) and belief propagation with ordered statistics decoding (OSD), highlighting trade-offs in accuracy and computational complexity. Simulation results under realistic noise models demonstrate the performance of different extraction and decoding schemes. The thesis concludes by emphasizing the importance of quantum error correction.