The properties of strongly correlated systems are of great interest but have often been challenging to elucidate. Some of these difficulties may be overcome by programmable digital quantum computers, which harness the quantum-mechanical nature of reality to simulate quantum systems and promise an advantage over computers rooted in classical physics. In this talk, I review developments in quantum algorithms advancing this goal, highlight the key role of physical insight such as the interaction picture and the finite speed of light driving recent progress, and point towards further challenges in quantum computation as a tool for fundamental physics.