As opposed to the standard (also called Schrödinger) approach where we keep a full vector of \(2^n\) amplitudes around for \(n\) qbits, in the sum-over-paths (also called Feynman) approach we focus on the final value of a single amplitude and trace all the different paths through the computation that contributed to it. Classically simulating quantum computers is of interest not only to complexity theorists, but anyone who cares about the promise of quantum computation!