<p>Among computational electromagnetic methods, Integral Equation (IE) solvers have a great capability in solving open-region problems such as scattering and radiation, due to no truncation boundary condition required. Volume Integral Equation (VIE) solvers are especially capable of handling arbitrarily shaped geometries and inhomogeneous materials. However, the numerical system resulting from a VIE analysis is a dense system, having $N^2$ nonzero elements for a problem of $ N $ unknowns. The dense numerical system in conjunction with the large number of unknowns resulting from a volume discretization prevents a practical use of the VIE for solving large-scale problems.</p> <p>In this work, two fast algorithms of $ O(N \log N) $ complexity to generate an rank-minimized $ H^2 $-representation for electrically large VIEs are developed. The algorithms systematically compress the off-diagonal admissible blocks of full VIE matrix into low-rank forms of total storage of $O(N)$. Both algorithms are based on nested cross approximation, which are purely algebraic. The first one is a two-stage algorithm. The second one is optimized to only use one-stage, and has a significant speedup. Numerical experiments on electrically large examples with over 33 million unknowns demonstrate the efficiency and accuracy of the proposed algorithms. </p> <p>Important applications of VIEs in thermal and quantum engineering have also been explored in this work. Creating spin(circularly)-polarized infrared thermal radiation source without an external magnetic field is important in science and engineering. Here we study two materials, magnetic Weyl semimetals and manganese-bismuth(MnBi), which both have permittivity tensors of large gyrotropy, and can emit circularly-polarized thermal radiations without an external magnetic field. We also design symmetry-broken metasurfaces, which show strong circularly-polarized radiations in simulations and experiments. In spin qubit quantum computing systems, metallic gates and antennas are necessary for quantum gate operations. But their existence greatly enhances evanescent wave Johnson noise (EWJN), which induces the decay of spin qubits and limits the quantum gate operation fidelity. Here we first use VIE to accurately simulate realistic quantum gate designs and quantify the influence on gate fidelity due to this noise.</p>