Quantum computing is presently in a “pre-vacuum-tube” era, with noisy devices that need error correction. But such architectures require many physical components, creating a high hardware and control overhead. Instead, a different paradigm based on individual devices, each with a high number of internal levels, has been proposed to achieve better hardware efficiency. The challenge then is how to accomplish universal control over such a multilevel system. This talk presents a new protocol for multi-level control (“matrix element modification”) in which the correspondence between control parameters and the resulting quantum dynamics of the oscillator is clear. This protocol was recently used to demonstrate high-dimensional quantum control of a harmonic oscillator, producing dynamics akin to that of a rotating top that are useful for logical quantum operations.