Chemical and molecular phenomena are central to understanding the composition of the physical universe. These mechanisms arise from quantum mechanical laws, which depict molecular systems as complex quantum states and can be rigorously investigated through first-principle calculations. Unfortunately, the most accurate attempts at these calculations are prohibitively costly from a computational perspective. In theory, some of these problems can be alleviated through a quantum computer, which utilizes a quantum mechanical state as its basic element. In practice, current-generation quantum computers are limited by noise, and so new algorithms and approaches have to be developed. The reduced density matrix allows us to efficiently characterize quantum states, as well as identify features of the obtained state. Additionally, these techniques provide insights into new algorithms and sources of error mitigation. In this work I explore the use of near-term (noisy, small to intermediate scale) quantum computers and the use of the two-electron reduced density matrix as a framework for tackling this problem. The importance of N-representability for error mitigation is seen from a number of perspectives. Additionally, a class of quantum contracted eigensolvers, which solve a contraction of the Schrodinger equation, are presented. These provide polynomially scaling approaches for highly accurate computations on near-term devices.