  The identification of materials with exceptional properties is an essential
objective to enable technological progress. We propose the application of
\textit{Quality-Diversity} algorithms to the field of crystal structure
prediction. The objective of these algorithms is to identify a diverse set of
high-performing solutions, which has been successful in a range of fields such
as robotics, architecture and aeronautical engineering. As these methods rely
on a high number of evaluations, we employ machine-learning surrogate models to
compute the interatomic potential and material properties that are used to
guide optimisation. Consequently, we also show the value of using neural
networks to model crystal properties and enable the identification of novel
composition--structure combinations. In this work, we specifically study the
application of the MAP-Elites algorithm to predict polymorphs of TiO$_2$. We
rediscover the known ground state, in addition to a set of other polymorphs
with distinct properties. We validate our method for C, SiO$_2$ and SiC
systems, where we show that the algorithm can uncover multiple local minima
with distinct electronic and mechanical properties.
