The accurate computation of Hamiltonian ground and excited states on quantum computers stands to impact many problems in the physical and computer sciences, from quantum simulation to machine learning. Given the challenges posed in constructing large-scale quantum computers, these tasks should be carried out in a resource-efficient way. We recently introduced  the quantum imaginary time evolution and quantum Lanczos algorithms, which are analogues of classical algorithms for finding ground and excited states. Compared with their classical counterparts, they require exponentially less space and time per iteration, and can be implemented without deep circuits and ancillae, or high-dimensional optimizations. We discuss applications to spins and fermions.  M. Motta, C. Sun, A. T. K. Tan, M. J. O'Rourke, E. Ye, A. J. Minnich, F. G. S. L. Brandao and G. K.-L. Chan, arXiv:1901.07653 *We acknowledge support by the US NSF, DOD and DOE. G. K.-L. Chan is a Simons Investigator in Physics and a member of the Simons Collaboration on the Many-Electron Problem. Hardware simulations were made possible by a grant through Rigetti Quantum Cloud Services, supported by the CQIA-Rigetti Partnership Program.