  Fundamental theories, such as Quantum Electrodynamics (QED) and Quantum
Chromodynamics (QCD) promise great predictive power addressing phenomena over
vast scales from the microscopic to cosmic scales. However, new
non-perturbative tools are required for physics to span from one scale to the
next. I outline recent theoretical and computational progress to build these
bridges and provide illustrative results for Hamiltonian Light Front Field
Theory. One key area is our development of basis function approaches that cast
the theory as a Hamiltonian matrix problem while preserving a maximal set of
symmetries. Regulating the theory with an external field that can be removed to
obtain the continuum limit offers additional possibilities as seen in an
application to the anomalous magnetic moment of the electron. Recent progress
capitalizes on algorithm and computer developments for setting up and solving
very large sparse matrix eigenvalue problems. Matrices with dimensions of 20
billion basis states are now solved on leadership-class computers for their
low-lying eigenstates and eigenfunctions.
