Multi-junction (“tandem”) solar cells have consistently achieved the top power-conversion efficiencies in the world for nearly three decades. In combination with solar concentrators and high-performance materials like GaAs, these devices routinely break the thermodynamic efficiency limit derived by Shockley and Queisser for single-junction devices. Here we model the limiting efficiency of a two-junction tandem solar cell with active layers of Perovskite material CH₃NH₃PbI₃ (MAPbI₃) and lead sulfide quantum dots (PbS QD). Starting from the ideal single-cell case (33.6 % maximum PCE), we then consider a version of the PVMirror system proposed by Yu et al., with a dichroic mirror splitting the solar spectrum between two ideal independently wired junctions. In the absence of solar concentration, such a system achieves a maximum power conversion efficiency of 45.7 %. We find that the optimal splitting wavelength (λ_split) for the ideal, independent tandem system always corresponds to the larger of the two bandgaps. Next we approximate the effects of non-ideal loss mechanisms by factoring in external quantum efficiency (EQE) data, resulting in maximum single-cell PCEs of 18.6 % and 23.4 % for 1.59-eV MAPbI₃ and the record PbS QD cell reported by Lan et al. The independent tandem system of the two materials achieves a maximum PCE of 25.6 % at λ_split = 610 nm. Preliminary constrained optimizations for tandem systems wired in series show a maximum PCE of 41.5 %. With further refinements to the algorithm, we aim to model a two-terminal monolithic device and account for the effect of solar concentration.