Adiabatic State Preparation
Reaching a hard-to-prepare ground state by starting easy and deforming slowly, guided by the adiabatic theorem.
The adiabatic theorem
The adiabatic theorem states that a system remaining in the instantaneous ground state of a slowly changing Hamiltonian will stay in the ground state, provided the change is slow compared with the inverse square of the spectral gap. This gives a recipe: begin in the easy ground state of a simple Hamiltonian H_start, then interpolate to the target H_target.
The interpolation
Define H(s) = (1 - s) H_start + s H_target for s going from 0 to 1, and evolve while ramping s slowly. If H_start has a trivially preparable ground state (say all spins aligned with a field) and the gap never closes along the path, the system ends in the ground state of H_target.
The gap governs the cost
- The required evolution time scales roughly as one over the minimum gap squared.
- A small gap (near a phase transition) forces slow, expensive evolution.
- Gap closing along the path can cause the method to fail (Landau-Zener transitions).
- Choosing a good interpolation path and schedule can keep the gap open.
Digital adiabatic preparation
On a gate-based machine, the continuous adiabatic sweep is Trotterized: the slowly-changing H(s) is simulated step by step. This makes adiabatic preparation a subroutine within digital simulation, often used to produce the input state for phase estimation, combining the physical intuition of adiabatic evolution with the rigor of the gate model.
Uses and limits
Adiabatic preparation is widely used to generate approximate ground states for chemistry and spin models, and it underlies adiabatic quantum computation and quantum annealing. Its Achilles' heel is the minimum gap: for hard instances the gap can be exponentially small, making the method exponentially slow, no better than the problem's intrinsic difficulty. Counterdiabatic driving and optimized schedules mitigate this in favorable cases. Adiabatic preparation is best seen as one tool among several for state preparation, strong when the gap stays open, unreliable when it does not.