The Adiabatic Theorem
A system stays in its instantaneous ground state if the Hamiltonian changes slowly enough relative to the energy gap; this is the principle behind adiabatic quantum computing and quantum annealing.
Slow change keeps you in the ground state
The adiabatic theorem says that if a quantum system starts in the ground state of a Hamiltonian H(t) and H(t) is varied slowly enough, the system tracks the instantaneous ground state throughout the evolution rather than being excited to higher levels.
How slow is slow enough
The required evolution time scales with the inverse square of the minimum spectral gap g_min between the ground and first excited states: T >> 1/g_min^2 (up to matrix-element factors). A large gap lets you move fast; a gap that nearly closes forces the schedule to crawl. The minimum gap along the path controls the whole cost.
Adiabatic quantum computing
- Encode the answer to a problem as the ground state of a final Hamiltonian H_P.
- Start in the easy-to-prepare ground state of a simple H_0.
- Interpolate H(s) = (1-s) H_0 + s H_P slowly from s=0 to s=1.
- If the interpolation is adiabatic, you end in the ground state of H_P -- the answer.
Why it matters, and its limit
Adiabatic quantum computing is polynomially equivalent to the circuit model, and its open-system heuristic cousin is quantum annealing. The catch is the gap: for hard (for example NP-hard) instances the minimum gap can shrink exponentially with system size, so the adiabatic runtime blows up. The gap, not the wall-clock schedule, is the real difficulty measure.