  Quantum materials exhibit a wide array of exotic phenomena and practically
useful properties. A better understanding of these materials can provide deeper
insights into fundamental physics in the quantum realm as well as advance
technology for entertainment, healthcare, and sustainability. The emergence of
digital quantum computers (DQCs), which can efficiently perform quantum
simulations that are otherwise intractable on classical computers, provides a
promising path forward for testing and analyzing the remarkable, and often
counter-intuitive, behavior of quantum materials. Equipped with these new
tools, scientists from diverse domains are racing towards achieving physical
quantum advantage (i.e., using a quantum computer to learn new physics with a
computation that cannot feasibly be run on any classical computer). The aim of
this review, therefore, is to provide a summary of progress made towards this
goal that is accessible to scientists across the physical sciences. We will
first review the available technology and algorithms, and detail the myriad
ways to represent materials on quantum computers. Next, we will showcase the
simulations that have been successfully performed on currently available DQCs,
emphasizing the variety of properties, both static and dynamic, that can be
studied with this nascent technology. Finally, we work through two examples of
how to map a materials problem onto a DQC, with full code included in the
Supplementary Material. It is our hope that this review can serve as an
organized overview of progress in the field for domain experts and an
accessible introduction to scientists in related fields interested in beginning
to perform their own simulations of quantum materials on DQCs.
