  Two objects can be distinguished if they have different measurable
properties. Thus, distinguishability depends on the Physics of the objects. In
considering graphs, we revisit the Ising model as a framework to define
physically meaningful spectral invariants. In this context, we introduce a
family of refinements of the classical spectrum and consider the quantum
partition function. We demonstrate that the energy spectrum of the quantum
Ising Hamiltonian is a stronger invariant than the classical one without
refinements. For the purpose of implementing the related physical systems, we
perform experiments on a programmable annealer with superconducting flux
technology. Departing from the paradigm of adiabatic computation, we take
advantage of a noisy evolution of the device to generate statistics of low
energy states. The graphs considered in the experiments have the same classical
partition functions, but different quantum spectra. The data obtained from the
annealer distinguish non-isomorphic graphs via information contained in the
classical refinements of the functions but not via the differences in the
quantum spectra.
