Colloquium: How to achieve quantum advantage?

The development of error-corrected quantum computers raises the prospect of quantum advantages in scientific computation. Which problems are likely to benefit, and what will it take to realize these advantages? I will first discuss basic quantum algorithm concepts and some criteria for comparing quantum and classical computation. I will then focus on eigenvalue problems, where improving the precision of an approximate solution offers one possible route to quantum advantage. Recent advances allow quantum eigensolvers to use few auxiliary qubits and short coherent evolutions, with near-optimal dependence on the target precision. I will introduce these developments and a method for estimating multiple eigenvalues together with their multiplicities. Most of the presentation is intended to be accessible to a broad mathematical audience without prior background in quantum computation.