Computable upper bounds for the adiabatic approximation errors

Abstract

For a given Hermitian Hamiltonian H(s) (s ∈ [0, 1]) with eigenvalues E _k (s) and the corresponding eigenstates |E _k (s)〉 (1 ⩽ k ⩽ N), adiabatic evolution described by the dilated Hamiltonian H _t (t):= H(t/T) (t ∈ [0, T]) starting from any fixed eigenstate |E _n (0)〉 is discussed in this paper. Under the gap-condition that |E _k (s) − E _n (s)| ⩾ λ > 0 for all s ∈ [0, 1] and all k ≠ n, computable upper bounds for the adiabatic approximation errors between the exact solution |ψ _T (t)〉 and the adiabatic approximation solution |ψ ^adi _T (t)〉 to the schrödinger equation $i\left| {\dot \psi _T \left. {(t)} \right\rangle = H_T (t)} \right|\left. {\psi _T (t)} \right\rangle$ with the initial condition |ψ _T (0)〉 = |E _n (0)〉 are given in terms of fidelity and distance, respectively. as an application, it is proved that when the total evolving time t goes to infinity, ‖|ψ _T (t)〉 − |ψ ^adi _T (t)〉‖ converges uniformly to zero, which implies that |ψ _T (t)〉 ≈ |ψ ^adi _T (t)〉 for all t ∈ [0, T] provided that T is large enough.