Recent experimental progress in laser technology has led to renewed interest in warm dense matter. Found in the interiors of gas giants and in inertial confinement fusion experiments, warm dense matter is relevant to problems of fundamental and technological importance but is a challenge to create experimentally and describe theoretically. Modern electronic structure theory, in the form of density functional theory coupled with molecular dynamics, in principle offers a route to describing realistic warm dense matter. However, until quite recently, no accurate exchange correlation free energy functionals existed and the accuracy of existing fits was largely unknown. %Moreover, existing accurate quantum Monte Carlo data for the exchange correlation energy of the warm dense electron gas differ substantially. In this thesis we extend the independent, systematically exact, density matrix quantum Monte Carlo method, to address these issues. Focussing on the warm dense uniform electron gas, we first outline how sampling issues present in the original formulation can be overcome and how numerical basis set corrections can significantly reduce the computational burden at high electronic temperatures. We next introduce a systematic approximation allowing larger system sizes to be tackled. In the process we resolve a controversy present between two competing path integral Monte Carlo methods, whose results for the exchange correlation energy of the uniform electron gas differ substantially in the warm dense regime. Finally, we develop a general procedure for deriving analytic finite size corrections in the warm dense regime, thus removing the final barrier to reaching the thermodynamic limit.