Collective-Excitation Spectra in Quantum Spin Glasses Out of Equilibrium

There exists an established connection between long-range interacting quantum spin glasses and combinatorial optimization problems, which serves as the main motivation behind the field of quantum annealing. On one hand, in disordered quantum spin systems, the focus is on exact methods (such as the replica trick) that allow the calculation of system quantities in the limit of infinite system and ensemble size. On the other hand, when solving a given instance of an optimization problem, disorder-averaged quantities are of no relevance, as one is solely interested in instance-specific, finite-size properties (such as the true solution). Here, we apply the Schwinger-Keldysh formalism to the semi-classical fluctuations around a spin mean-field [PRX Quantum extbf{4}, 030335 (2023)] to obtain the excitation spectrum of the non-equilibrium dynamics along the annealing path. For the example of the quantum Sherrington-Kirkpatrick spin glass, we show that this method provides approximate access to the instance-specific location (along the path) and size of the minimal gap, which is a quantity crucial to quantum annealing.