  The estimation of low energies of many-body systems is a cornerstone of
computational quantum sciences. Variational quantum algorithms can be used to
prepare ground states on pre-fault-tolerant quantum processors, but their lack
of convergence guarantees and impractical number of cost function estimations
prevent systematic scaling of experiments to large systems. Alternatives to
variational approaches are needed for large-scale experiments on
pre-fault-tolerant devices. Here, we use a superconducting quantum processor to
compute eigenenergies of quantum many-body systems on two-dimensional lattices
of up to 56 sites, using the Krylov quantum diagonalization algorithm, an
analog of the well-known classical diagonalization technique. We construct
subspaces of the many-body Hilbert space using Trotterized unitary evolutions
executed on the quantum processor, and classically diagonalize many-body
interacting Hamiltonians within those subspaces. These experiments show that
quantum diagonalization algorithms are poised to complement their classical
counterpart at the foundation of computational methods for quantum systems.
