Stabiliser operations and state preparations are efficiently simulable by classical computers. Stabiliser circuits play a key role in quantum error correction and fault-tolerance, and can be promoted to universal quantum computation by the addition of "magic" resource states or non-Clifford gates. It is believed that classically simulating stabiliser circuits supplemented by magic must incur a performance overhead scaling exponentially with the amount of magic. Early simulation methods were limited to circuits with very few Clifford gates, but the need to simulate larger quantum circuits has motivated the development of new methods with reduced overhead. A common theme is that algorithm performance can often be linked to quantifiers of computational resource known as magic monotones. Previous methods have typically been restricted to specific types of circuit, such as unitary or gadgetised circuits. In this thesis we develop a framework for quantifying the resourcefulness of general qubit quantum circuits, and present improved classical simulation methods. We first introduce a family of magic state monotones that reveal a previously unknown formal connection between stabiliser rank and quasiprobability methods. We extend this family by presenting channel monotones that measure the magic of general qubit quantum operations. Next, we introduce a suite of classical algorithms for simulating quantum circuits, which improve on and extend previous methods. Each classical simulator has performance quantified by a related resource measure. We extend the stabiliser rank simulation method to admit mixed states and noisy operations, and refine a previously known sparsification method to yield improved performance. We present a generalisation of quasiprobability sampling techniques with significantly reduced exponential scaling. Finally, we evaluate the simulation cost per use for practically relevant quantum operations, and illustrate how to use our framework to realistically estimate resource costs for particular ideal or noisy quantum circuit instances
