Today the availability and the fidelity of real quantum hardware are limited as well as the information that can be directly retrieved from experiments on it. To fully explore the possibilities that quantum algorithms can offer and to reliably estimate the performance of a real quantum computer in a practical scenario, a classical quantum circuit simulator is needed. Following the Dirac notation: noiseless and noisy quantum states and their evolution under quantum gates and non-ideality phenomena, such as decoherence, are described by vectors and matrices composed of complex numbers. The main problem related to this formalism is the exponential increase of both simulation time and memory occupation with the number of qubits.