  Experimental design techniques such as active search and Bayesian
optimization are widely used in the natural sciences for data collection and
discovery. However, existing techniques tend to favor exploitation over
exploration of the search space, which causes them to get stuck in local
optima. This ``collapse" problem prevents experimental design algorithms from
yielding diverse high-quality data. In this paper, we extend the Vendi scores
-- a family of interpretable similarity-based diversity metrics -- to account
for quality. We then leverage these quality-weighted Vendi scores to tackle
experimental design problems across various applications, including drug
discovery, materials discovery, and reinforcement learning. We found that
quality-weighted Vendi scores allow us to construct policies for experimental
design that flexibly balance quality and diversity, and ultimately assemble
rich and diverse sets of high-performing data points. Our algorithms led to a
70%-170% increase in the number of effective discoveries compared to baselines.
