Diabatic Transitions

The adiabatic theorem only guarantees success if the evolution is slow enough. When it is not, the system jumps out of the instantaneous ground state into an excited state — a diabatic transition — and the final measurement returns the wrong answer. This lesson explains where these transitions come from and quantifies them with the Landau–Zener formula.

Avoided crossings

As the schedule parameter $s$ sweeps from 0 to 1, the energy levels $E_0(s)$ and $E_1(s)$ move. At an avoided crossing they approach closely — separated only by the minimum gap $\Delta_{\min}$ — and then move apart again. Near such a point the ground and first-excited states are nearly degenerate, and even a small perturbation (a slightly-too-fast schedule, a stray coupling) can mix them. The smaller the gap, the easier it is to leak amplitude into the excited state.

If the gap actually closed — a true level crossing with $\Delta = 0$ — the adiabatic theorem would fail outright: there would be no way to distinguish the two states, and the system would generically split between them. Avoided crossings are the dangerous-but-survivable version of this.

The Landau–Zener formula

Model the two relevant levels near a crossing as a two-state system whose gap passes through its minimum $\Delta_{\min}$ while the energy bias is swept linearly in time at rate $\alpha$ (the speed at which the diagonal energy difference changes). The probability of a diabatic transition — staying on the original diabatic branch instead of following the ground state — is

P_{\text{diabatic}} = \exp\!\left(-\frac{\pi\,\Delta_{\min}^2}{2\hbar\,\alpha}\right).

Read the formula carefully:

Larger gap $\Delta_{\min}$ exponentially suppresses the transition — wide gaps are safe.
Faster sweep (larger $\alpha$ ) exponentially enhances the transition — speed kills.
Staying adiabatic means $P_{\text{diabatic}} \to 0$ , which requires $\alpha \ll \Delta_{\min}^2/\hbar$ , i.e. a runtime $T \sim 1/\alpha \gg 1/\Delta_{\min}^2$ — the same scaling the adiabatic theorem predicts.

Consequences for annealing

In a real anneal several effects conspire to cause diabatic errors:

Small gaps from hard instances. When $\Delta_{\min}$ is exponentially small, no realistic schedule is slow enough; the anneal almost certainly ends in an excited state.
Finite coherence time. Hardware cannot run arbitrarily long, so very slow schedules are not an option — there is a tension between adiabaticity and decoherence.
Thermal excitation. At nonzero temperature the environment can also kick the system across the gap, independent of schedule speed.

Practitioners fight back with pausing (dwelling near the minimum gap to re-thermalize toward the ground state), reverse annealing (starting from a candidate solution), and schedule shaping that slows down precisely where the gap is smallest. Interestingly, deliberately controlled diabatic transitions are sometimes useful: "diabatic quantum annealing" exploits fast passages through excited states to reach the target faster than a strictly adiabatic path.