Abstract

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.
Introduction.—Kak’s [21] question “Can one hear the shape of a drum?” is part of the scientific pop culture [30]. The technical side of the question concerns our ability to completely specify the geometry of a domain from the eigenvalues of its Laplacian. The question has been reinterpreted in the study of Schrödinger operators on metric graphs by Gutkin and Smilansky [15] and restated in Algebraic Graphs Theory as “Which graphs are determined by their spectrum?” by van Dam and Haemers [9].(Through this work, the …

Date

January 1, 1970

Authors

Walter Vinci, Klas Markström, Sergio Boixo, Aidan Roy, Federico M Spedalieri, Paul A Warburton, Simone Severini

Journal

arXiv preprint arXiv:1307.1114