Continuous-time quantum walks on exponentially large, sparse graphs form a powerful paradigm for quantum computing: On the one hand, they can be efficiently simulated on a quantum computer. On the other hand, there are BQP-complete problems based on continuous-time quantum walks, which provides an alternative framework for thinking about quantum computing - a perspective which has indeed led to a number of novel algorithms and oracle problems. Recently, simulating the dynamics of a system of harmonic oscillators (that is, masses and springs) was set forth as another BQP-complete problem defined on exponentially large, sparse graphs. In this work, we establish a mapping between these two classes of problems. As compared to linking the two classes of problems via their BQP-completeness, our mapping has several desirable features: it preserves the structure of the problem, including initialization, read-out, and efficient oracle access, resulting in low overhead in terms of both space and time. It also naturally preserves the geometry of the underlying graph and provides a recipe to translate any quantum algorithm in the quantum walk paradigm to harmonic oscillators (and vice versa), which allows one to map between restricted subsets of instances that are not BQP-complete, and offers an alternative proof for the BQP-completeness of the harmonic oscillator problem via a reduction to and from the BQP-complete quantum walk problem.