Self-Healing of Trotter Error in Digital Adiabatic State Preparation

Adiabatic time evolution can be used to prepare a complicated quantum many-body state from one that is easier to synthesize and Trotterization can be used to implement such an evolution digitally. The complex interplay between nonadiabaticity and digitization influences the infidelity of this process. We prove that the first-order Trotterization of a complete adiabatic evolution has a cumulative infidelity that scales as $\mathcal{O}({T}^{\ensuremath{-}2}\ensuremath{\delta}{t}^{2})$ instead of $\mathcal{O}({T}^{2}\ensuremath{\delta}{t}^{2})$ expected from general Trotter error bounds, where $\ensuremath{\delta}t$ is the time step and $T$ is the total time. This result suggests a self-healing mechanism and explains why, despite increasing $T$, infidelities for fixed-$\ensuremath{\delta}t$ digitized evolutions still decrease for a wide variety of Hamiltonians. It also establishes a correspondence between the quantum approximate optimization algorithm and digitized quantum annealing.