  We present a novel method for solving eigenvalue problems on a quantum
computer based on spectroscopy. The method works by coupling a "probe" qubit to
a set of system simulation qubits and then time evolving both the probe and the
system under Hamiltonian dynamics. In this way, we simulate spectroscopy on a
quantum computer. We test our method on the IBM quantum hardware for a simple
single spin model and an interacting Kitaev chain model. For the Kitaev chain,
we trace out the pseudo-topological phase boundary for a two-site model.
