  The last five years have seen a dramatic evolution of platforms for quantum
computing, taking the field from physics experiments to quantum hardware and
software engineering. Nevertheless, despite this progress of quantum
processors, the field is still in the noisy intermediate-scale quantum (NISQ)
regime, seriously limiting the performance of software applications. Key issues
involve how to achieve quantum advantage in useful applications for quantum
optimization and materials science, connected to the concept of quantum
supremacy first demonstrated by Google in 2019. In this article we will
describe recent work to establish relevant benchmarks for quantum supremacy and
quantum advantage, present recent work on applications of variational quantum
algorithms for optimization and electronic structure determination, discuss how
to achieve practical quantum advantage, and finally outline current work and
ideas about how to scale up to competitive quantum systems.
