The Adiabatic Theorem

The adiabatic theorem is the physical principle behind a whole model of quantum computation. Informally it says: if a quantum system starts in the ground state of a Hamiltonian and that Hamiltonian is changed slowly enough, the system stays in the instantaneous ground state throughout the change. "Slowly enough" is made precise by the energy spectrum — specifically by the gap to the first excited state.

Time-dependent Hamiltonians

A closed quantum system evolves under the Schrödinger equation

i\hbar \frac{d}{dt}\,|\psi(t)\rangle = H(t)\,|\psi(t)\rangle .

When $H$ does not depend on time, the energy eigenstates are stationary. Adiabatic computation deliberately uses a Hamiltonian that does depend on time, interpolating from an easy starting Hamiltonian to one whose ground state encodes the answer to a problem.

At every instant $t$ the operator $H(t)$ has eigenstates and eigenvalues

H(t)\,|n; t\rangle = E_n(t)\,|n; t\rangle, \qquad E_0(t) \le E_1(t) \le E_2(t) \le \cdots

We label the instantaneous ground state $|0; t\rangle$ . The quantity that controls everything is the spectral gap

\Delta(t) = E_1(t) - E_0(t) \ge 0 ,

the energy difference between the ground state and the first excited state at time $t$ .

The statement

Suppose $H(t)$ varies smoothly over a total time $T$ , written through a normalized schedule $s = t/T \in [0,1]$ so that $H(t) = \tilde{H}(s)$ . If the system begins in the ground state $|0; 0\rangle$ and the gap never closes, the adiabatic theorem guarantees that the final state is close to the instantaneous ground state $|0; 1\rangle$ , provided $T$ is large enough.

A standard sufficient condition is

T \;\gg\; \frac{\max_{s}\, \bigl\| \partial_s \tilde{H}(s) \bigr\|}{\min_{s}\, \Delta(s)^2}.

Two features deserve emphasis:

The minimum gap $\Delta_{\min} = \min_{s} \Delta(s)$ enters as $\Delta_{\min}^{2}$ in the denominator. A small gap forces a long runtime; the required time scales like $1/\Delta_{\min}^2$ .
The numerator measures how fast the Hamiltonian itself changes. Spreading the same change over more time (larger $T$ ) makes the evolution gentler.

Why staying in the ground state is useful

If you can engineer a final Hamiltonian whose ground state encodes the solution to a computational problem, and an initial Hamiltonian whose ground state is easy to prepare, then adiabatic evolution transports you from the easy state to the answer. The art is choosing the interpolation so the gap stays open and the runtime stays manageable. Later lessons build exactly such Hamiltonians, design schedules, and quantify the cost through the gap.

What can go wrong

If the Hamiltonian changes too quickly relative to $\Delta_{\min}^2$ , the system can be excited out of the ground state — a diabatic transition. The probability of leaking into the first excited state grows as the evolution gets faster or the gap gets smaller. Avoiding such transitions, by slowing down near the minimum gap, is the central practical challenge of adiabatic and annealing-based computing.