I will describe two lines of work where my mathematical background plays a central role. I will begin with earlier work on non-Abelian vortex knots, where diagrammatic methods make it possible to visualize, classify, and reason about the stability of three-dimensional structures with respect to local reconnections and vortex crossings. I will conclude with my current work at QMill, a quantum-algorithms startup focused on extracting near-term value from noisy intermediate-scale quantum (NISQ) hardware, where we have been making substantial recent progress.