Abstract:
Quantum computing becomes achievable only assuming that fault-tolerant quantum error correction codes (QECCs) can be effectively implemented on physical hardware, offering significantly lower error rates than those inherent in the system. Quantum stabilizer codes encompass various families of quantum codes and furnish a group-theoretic framework for quantum error correction (QEC). Each code is distinguished by a pseudo-threshold value, indicative of its performance under a selected noise model and decoding scheme. We establish a framework capable of generating error correction circuits and providing error rates and pseudo-thresholds, if applicable, for any stabilizer code. Additionally, we introduce a unitary encoding scheme capable of encoding any stabilizer code, overcoming the limitation of disregarding the phase of a stabilizer generator. Furthermore, we devise a syndrome extraction circuit employing a single ancilla qubit, which accelerates simulation while minimizing the consumption of qubit resources. Fault tolerance stands as a crucial requirement that renders QECCs practically useful. Among the myriad techniques to achieve fault tolerance, we concentrate on the bare ancillary fault tolerance scheme, which boasts the least overhead observed to date. Our focus involves a review of the reordering stabilizer generator trick for syndrome extraction, with an extension aimed at overcoming limitations inherent in the original method proposed by Brown et. al. Furthermore, we delve into a study of the fault-tolerant properties exhibited by an eight-qubit code, obtaining pseudo-threshold values under the standard depolarizing and anisotropic noise models. Additionally, we undertake a comparative analysis of various error rates under the bare ancillary fault tolerance method, elucidating why a noise-free error correction step is often employed in the literature despite its practical infeasibility. Our work offers a comprehensive introduction to fault-tolerant QEC and extends it to the simulation of quantum error-correcting codes. The techniques outlined in this thesis complement existing literature and surmount limitations to achieve lower error rates in physical systems.
