The adiabatic theorem says that a quantum system started in an eigenstate of its Hamiltonian stays in the corresponding eigenstate as the Hamiltonian changes, provided the change is slow enough and the level never touches its neighbours. It is the formal version of an idea that is easy to state and easy to get wrong: go slowly and the system keeps up. What counts as slow is set by the energy gap, and that turns out to be the whole story.

Let $H(t)$ have instantaneous eigenstates and eigenvalues

$$H(t)\,\lvert n(t)\rangle = E_n(t)\,\lvert n(t)\rangle \tag{1.1}$$

These are defined pointwise in time: at each instant, freeze the Hamiltonian and diagonalise it. Nothing about $\lvert n(t)\rangle$ requires the system to actually be in that state.

If the system starts in $\lvert\psi(0)\rangle = \lvert n(0)\rangle$ and $H$ varies slowly, then at all later times

$$\lvert\psi(t)\rangle \;\approx\; e^{i\theta_n(t)}\,e^{i\gamma_n(t)}\,\lvert n(t)\rangle$$

with two phases out front and nothing else. The state is still the $n$-th eigenstate, just of a different Hamiltonian than it started with.

The first phase is the dynamical phase, the one that would be there even if $H$ never changed:

$$\theta_n(t) = -\frac{1}{\hbar}\int_0^t E_n(t')\,dt'$$

The second is the geometric or Berry phase, which depends on the path taken through parameter space and not on how long the trip took:

$$\gamma_n(t) = i\int_0^t \big\langle n(t') \big\rvert \partial_{t'} n(t')\big\rangle\,dt'$$

Slowly compared to what is the only question worth asking, and the answer is not “compared to the total run time”. Expand the state in the instantaneous basis, $\lvert\psi\rangle = \sum_n c_n e^{i\theta_n}e^{i\gamma_n}\lvert n\rangle$, feed it to the Schrödinger equation, and the coefficients obey

$$\dot c_m = -\sum_{n \neq m} c_n \,\big\langle m \big\rvert \partial_t n \big\rangle\, e^{i(\theta_n - \theta_m)}e^{i(\gamma_n - \gamma_m)}$$

The leakage out of level $m$ is controlled entirely by $\langle m \rvert \partial_t n\rangle$, and differentiating the eigenvalue equation turns that into something measurable:

$$\big\langle m \big\rvert \partial_t n \big\rangle = \frac{\big\langle m \big\rvert \partial_t H \big\rvert n \big\rangle}{E_n - E_m}, \qquad m \neq n$$

There is the gap, in the denominator, where it stays for the rest of the subject. The standard adiabatic condition follows:

$$\hbar\,\frac{\big| \big\langle m \big\rvert \partial_t H \big\rvert n \big\rangle \big|}{\big(E_m - E_n\big)^2} \;\ll\; 1$$

Note the square. Halving the gap does not double the time needed, it quadruples it. This single fact is why gap estimates dominate every practical discussion of adiabatic methods, and why a problem with an exponentially small gap is not slow but hopeless.

The dashed lines are the states that would have crossed. The solid ones are the true eigenvalues, which repel and leave a gap of $\Delta_{\min}$. Adiabatic evolution is the claim that a system on the lower solid branch stays on it, going in one side and coming out the other having quietly swapped its character.

The cleanest model is two levels sweeping through an avoided crossing:

$$H(t) = \begin{pmatrix} v t & \Delta \\ \Delta & -v t \end{pmatrix}, \qquad E_\pm(t) = \pm\sqrt{v^2t^2 + \Delta^2}$$

The gap is smallest at $t = 0$, where it equals $2\Delta$, and the sweep rate $v$ is what the system is being asked to keep up with. Below, the left panel is the spectrum with the state's position on it, and the right panel is the overlap $|\langle 0(t) \lvert \psi(t)\rangle|^2$ with the instantaneous ground state. The Schrödinger equation is integrated live rather than being drawn from a formula.

Each pass raises the sweep rate and then starts over, and the reset button takes it back to the slowest pass. At small $v$ the green curve dips at the crossing and comes back to one. At large $v$ it dips and stays down: the system went straight through, keeping its old character rather than its old energy level, and the run has failed.

For this model the leaked probability is known exactly and in closed form, which is unusual and useful. The probability of not following the adiabatic branch is

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

where $\Delta_{\min}$ is the minimum gap and $\alpha$ is the rate at which the two diabatic levels separate. The shape of that expression is the practical content of the whole theorem:

The exponential is the point. Success is not linear in patience: below a threshold set by the gap you get essentially perfect transfer, and above it you fall off a cliff. Doubling the run time of a failing schedule usually does nothing at all.

The oldest version of the theorem is a spin in a magnetic field that slowly changes direction. The spin precesses about $\vec B$ at the Larmor frequency, and if $\vec B$ turns much more slowly than that, the spin's cone of precession is dragged along with it.

Both panels below start with the spin aligned to the field. Only the rotation rate differs. The right panel wanders further the longer it runs, so this one carries a reset button as well: it reloads the sketch and puts both spins back on the field.

The grey vector is the field and the orange one is the spin. On the left, $\vec B \cdot \vec r$ stays pinned near one: the spin is still aligned, which is the adiabatic theorem doing its job. On the right the alignment wanders over the whole range, because the field moved out from under the precession before it could be dragged.

The dynamical phase is unsurprising: it accumulates energy times time. The geometric phase is the interesting one, because it survives the limit of infinitely slow evolution. Take the parameters around a closed loop and the dynamical phase depends on how long you dawdled, while

$$\gamma_n = i\oint_{\mathcal C} \big\langle n(\vec R) \big\rvert \nabla_{\vec R} n(\vec R)\big\rangle \cdot d\vec R$$

does not. It depends on the loop and nothing else. For a spin one-half in a field of fixed magnitude, the answer is famously just geometry:

$$\gamma_\pm = \mp\tfrac{1}{2}\,\Omega$$

with $\Omega$ the solid angle the field's direction traced out on the sphere. This is the same bundle geometry as the Hopf fibration: the loop lives in the base, the phase lives in the fiber, and the connection is what relates them.

The theorem stops being a curiosity about slow spins the moment someone notices it is a computational model. Encode a problem so that its answer is the ground state of a Hamiltonian $H_1$ nobody knows how to prepare, start instead in the easy ground state of some $H_0$, and interpolate:

$$H(s) = (1-s)\,H_0 + s\,H_1, \qquad s = t/T \in [0,1]$$

Prepare the ground state of $H_0$, run $s$ from $0$ to $1$ slowly enough, and the adiabatic theorem hands over the ground state of $H_1$, which is the answer. This is equivalent in power to the circuit model, not a lesser relative of it, which is not obvious and was not proved until 2004.

The catch is the same square as before. The run time is set by the smallest gap encountered anywhere along the path:

$$T \;\gtrsim\; \frac{\max_s \big| \big\langle 1(s) \big\rvert \partial_s H \big\rvert 0(s)\big\rangle \big|}{\min_s\, g(s)^2}, \qquad g(s) = E_1(s) - E_0(s)$$

So the entire difficulty of the model is pushed into one question, and it is a question about the spectrum rather than about the algorithm:

A gap that shrinks polynomially in the problem size gives a polynomial run time. A gap that shrinks exponentially gives an exponential one, and no amount of engineering rescues it. Proving which case a given problem falls into is, in general, exactly as hard as the problem.

Quantum annealing is the practical cousin: the same interpolation, run on hardware, at finite temperature, faster than the theorem strictly allows, and judged on whether the answers are good rather than on whether the premises hold. The adiabatic theorem is where its intuition comes from, not a description of what it does.

Worth knowing, because each of these breaks it in a different way.

The first row deserves emphasis. The theorem does not say that slow evolution is safe. It says that slow evolution is safe when the level stays isolated, and the whole practical art is in knowing whether it does.

• Schrödinger equation
• Eigenstate
• Hopf fibration
• Bloch sphere
• Global phase