High-energy-density (HED) science concerns the physics and chemistry of matter under extreme conditions, where pressure ranges from millions up to trillions of atmospheres. Experimental and computational investigations of the quantum nature of HED matter have revealed many remarkable phenomena, including pressure-induced superconductivity, diamond precipitation in carbon-bearing compounds, and the emergence of an entirely new high-pressure periodic table of the elements. A deeper understanding of quantum HED matter has far-reaching implications for planetary science, astrophysics, and technological applications, including harnessing clean energy through inertial confinement fusion and designing novel quantum materials. Computational studies using first-principles and ab initio methods play a vital role in advancing this emerging field. This book provides a detailed introduction to these quantum-mechanical methods essential for the computational study of HED science. It outlines practical procedures for calculating key material properties of quantum HED matter, including equation of state, transport coefficients, and radiative properties. Designed as a reference book, it offers fundamental knowledge of computational HED physics for graduate students, postdoctoral researchers, and early-career scientists entering this rapidly evolving field.