The computation of loop scattering amplitudes is a crucial task that must be undertaken to perform precise predictions of Standard Model processes at collider experiments. In this talk, I will overview the problems encountered when performing two-loop amplitude calculation and the state-of-the-art technologies that we employ to tackle them. Thereafter, I will discuss a proposal for breaking a bottleneck in the computation of amplitudes depending on an increasing number of particles/heavy fields. The approach applies tools from algebraic geometry in order to construct compact templates (Ansätze) for the amplitudes, thereby enabling efficient computation.