With this thesis project, we improve the classical simulation of quantum computers using stabilizers in the GSLC formalism. We do this in two ways: we present new algorithms that speed up their simulation and extend their applicability by defining new operations and subroutines for existing general circuit simulation using GSLC. To be precise: we present multiple new algorithms that speed up the simulation of the CZ gates, the most computationally expensive quantum operation in GSLC formalism. We define two new operations on GSLC that are useful when simulating stabilizer circuits: calculating fidelity (a measure of 'closeness' between two quantum states), and tracing out qubits (throwing away the information about the state contained in these qubits) from a GSLC. Finally, we present a new GSLC-based subroutine for a state of the art general quantum circuit simulation algorithm by Bravyi et al. that allows for the usage of the faster CZ algorithms. We show that the GSLC formalism can give a speedup in practical simulation tasks by evaluating the complexity of simulating an algorithm with possible applications on near-term quantum hardware: the quantum approximate optimization algorithm.