Open quantum systems that do not have the Markovian property are generally much more difficult to solve. This is largely due to the fact that the next state of a non-Markovian system is determined by each of its previous states, which rapidly increases the memory requirements to compute the evolution of the system. Currently, the methods of treating these systems employ what are known as projection operator techniques. These techniques employ a projection operator P , which effectively applies the trace over the environment as described previously. The result of applying P to ρ ρ (i.e. calculating P ρ ρ ) is called the relevant part of ρ ρ . For completeness, another operator Q is defined so that P + Q = I where I is the identity matrix. The result of applying Q to ρ ρ (i.e. calculating Q ρ ρ ) is called the irrelevant part of ρ ρ . The primary goal of these methods is to then derive a master equation that defines the evolution of P ρ ρ .