Achieving universal and scalable quantum computing with reliably low error rates, despite the presence of unreliable circuit components, requires fault-tolerant quantum error correction. In general, quantum error correction imposes a significant overhead on the computation, motivating exploration of opportunities for optimization. Flag fault tolerance protocols have emerged as important schemes to realize fault tolerance experiments in the near term, because of their low qubit overhead, and absence of strict requirement for elaborate ancillary state preparation, relative to traditional schemes. However, the existing fast-reset, single-flag protocols for small codes generally employ a measurement of all stabilizer generators with unflagged circuits to distinguish a limited set of errors via the syndrome, leading to high circuit depth. In addition, the flagged measurement outcomes play a limited role in differentiating these errors. This motivates the possibility of reducing the circuit depth fault-tolerantly in flag-based syndrome extraction circuits. In this thesis, flag protocols with significantly reduced number of stabilizer measurements are constructed for the [[5,1,3]] code and the Steane code. The new protocols are divided into two classes. In the first class, the reduction is achieved by a dynamic choice of unflagged stabilizer measurements, based on past syndromes, and the utilization of the complete stabilizer group, to distinguish restricted sets of errors signalled by respective flagged measurements. In the second class, the reduction is achieved by measuring three high-weight flagged stabilizers, with the capability to detect a single input error, for the Steane code. The reduced stabilizer sequences are methodically constructed to yield unique and nontrivial syndromes for the relevant error set. This ensures that the fundamental condition of errors being detectable and distinguishable, which is the principal factor for the existing flag protocols to be fault-tolerant, is preserved. Pseudothresholds competitive with the existing flag protocols are established via Monte Carlo simulations under an error model consisting of two-qubit gate depolarizing errors, state preparation errors and measurement errors. Additionally, computer search programs are developed to obtain analogous reduced stabilizer sequences for both classes. These programs are also employed to assist in identifying certain mathematical properties of the high-weight Steane code stabilizers which can detect a single input error: namely, these stabilizers belong to different cosets of the X-stabilizer subgroup, and arise from 8-element subgroups within the stabilizer group. Furthermore, examples of such stabilizer sequences are constructed for few other codes. This thesis highlights the potential of employing parity measurements from the complete stabilizer group and extending beyond conventional adaptive measurements to improve the resource efficiency of fault-tolerant quantum error correction.

This thesis presents a novel formulation to study the qubit-mapping problem (QMP). The presented for- mulation redefines the problem in terms of density matrices which represent the quantum algorithm and the underlying architecture—allowing the implementation of techniques from quantum information theory to es- tablish a bounded metric space for comparing these density matrices. The main contribution of this thesis is implementing this formulation in an algorithm to determine the minimal bound on the required number of SWAP operations for a pairing of a quantum algorithm to an underlying device where the initial mapping has been provided. Benchmarks have shown a clear dependence on the β-value. Emphasising the need for future investigations of this dependence to enhance the algorithm’s effectiveness for more extensive algorithms and architectures. While it is essential to acknowledge that the approach may not currently rival the state of the art.