Self-healing of Trotter error in digital adiabatic state preparation

Abstract

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 non-adiabaticity 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 $\mathcal O(T^{-2} \delta t^2)$ instead of $\mathcal O(T^2 \delta t^2)$ expected from general Trotter error bounds, where $\delta t$ 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-$\delta t$ digitized evolutions still decrease for a wide variety of Hamiltonians. It also establishes a correspondence between the Quantum Approximate Optimization Algorithm (QAOA) and digitized quantum annealing.