Adiabatic preparation of many-body quantum states: getting the beginning and ending right

We present numerical calculations, and simulations performed on a Rydberg atom quantum simulator, of the adiabatic evolution of many-body quantum systems around a quantum phase transition. We demonstrate that the end-to-end transfer error, for a given process duration and dissipative losses, can be suppressed by adopting smooth initial and final scheduling functions for the Hamiltonian. We consider a one-dimensional mixed-field Ising model, as well as a chain of Rydberg atoms, and compare numerical calculations and experimental results for the end-to-end transfer error with different schedule functions. We show, in particular, that if the time dependent Hamiltonian is $n$ times differentiable with vanishing $1^{st}$ to $n^{th}$ order derivatives in the beginning and in the end, the infidelity with respect to the final adiabatic eigenstate scales as $1/T^{n+1}$ when evolving for time $T$.

💡 Research Summary

The paper investigates how the smoothness of the Hamiltonian at the beginning and end of an adiabatic protocol influences the final preparation error of many‑body quantum states. The authors first prove a new adiabatic theorem: if the time‑dependent Hamiltonian (H(\tau)) (with (\tau = t/T)) is (n)-times continuously differentiable and its first (n) derivatives vanish at both (\tau=0) and (\tau=1), then the distance between the evolved state and the target instantaneous eigenstate scales as (O(\epsilon^{,n+1})) with (\epsilon = 1/T). This improves on earlier rigorous bounds of (O(\epsilon^{,n})) by one order, and the proof relies on a super‑adiabatic expansion where the boundary term dominates the leading error. The theorem identifies two regimes: for sufficiently small (\epsilon) (the “polynomial regime”) the error follows the (\epsilon^{,n+1}) power law, while for larger (\epsilon) the error decays exponentially as (\exp(-c/\epsilon)), with the constant (c) set by analyticity properties of (H(\tau)) and the minimal spectral gap.

To test the theorem, the authors study two concrete platforms. The first is a one‑dimensional mixed‑field Ising chain \