Recent years have seen tremendous progress toward building programmable quantum computers, with a variety of private companies and government agencies, as well as academic institutions, investing heavily in the emerging technology. Quantum computers promise to solve certain problems that are intractable for classical computers, but the utility of existing devices is severely restricted by noise. This thesis details several attempts to overcome this noise and make use of existing devices to address outstanding questions in physics. The work presented here demonstrates that, by employing error mitigation and avoidance techniques, existing devices can be used to study both ground state physics (Chapters 2, 3) and dynamics (Chapters 5, 6) of local spin systems. I also consider a non-local system of Majorana fermions (Chapter 4) but find its simulation to be beyond the capabilities of existing quantum processors at nontrivial system sizes. Whether the successful quantum simulations detailed here are extensible beyond the reach of classical techniques remains an open question, but the rapid pace of development suggests that useful quantum advantage may occur soon. Of the quantum simulations presented here, that of Chapter 5 is the most promising candidate for useful quantum advantage because it both addresses an outstanding question in the scientific community and extends beyond our classical simulations. However, more work is needed to determine whether it is beyond the capabilities of approximate classical methods as well. I expect that, in the coming years, quantum processors will be able to perform quantum simulations that are increasingly useful for the scientific community and increasingly difficult to simulate classically.