Adiabatic theorem

If a Hamiltonian changes sufficiently slowly while an eigenstate remains separated from the rest of the spectrum by a nonzero gap, evolution keeps the state in the corresponding instantaneous eigenspace up to a dynamical and geometric phase. A degenerate isolated eigenspace can undergo a unitary holonomy within itself.

Expanding a quantum state in instantaneous energy eigenstates separates dynamical phases from the coupling $A_{mn}$. orthonormality makes $A$ anti-Hermitian. For distinct eigenvalues, differentiating the eigenvalue equation gives $A_{mn}=⟨m|\dot{H}|n⟩/(E_n-E_m)$. Its off-diagonal entries drive nonadiabatic transitions; diagonal entries describe phase changes of the basis and generate the Berry phase.

A Hamiltonian path preserving the history subspace can be analysed using its restricted spectral gap. If this gap is $\Omega (T^{-2})$ and the derivative has operator norm $O(1)$, a runtime criterion proportional to $\Vert \dot{H}{\Vert }^2/(ϵ{\Delta }^3)$ permits computational history state preparation in time $O(T^6/ϵ)$. Other invariant sectors do not cause leakage from a state initialized in the history subspace.