Subatomic particles known as hadrons, such as protons and neutrons, are composite objects comprised of elementary particles called quarks that interact via the exchange of massless gluons. The dynamics of quarks and gluons inside hadrons are described by a theory known as quantum chromodynamics. To date, the best approach to calculating hadron properties is to formulate this theory on a space-time lattice and apply Monte Carlo integration methods. Improving Monte Carlo algorithms and the use of high-performance computing resources of increasing power have led to significant progress in understanding the lightest hadrons from this theory. This work describes a recent algorithmic advance that has opened up new studies of large numbers of heavier excited hadrons. The availability of petascale heterogeneous computing systems, such as those provided by NSF XSEDE, is crucial to the success of such studies.