Adiabatic rotation, quantum search, and preparation of superposition states

We introduce the idea of using adiabatic rotation to generate superpositions of a large class of quantum states. For quantum computing this is an interesting alternative to the well-studied “straight line” adiabatic evolution. In ways that complement recent results, we show how to efficiently prepare three types of states: Kitaev’s toric code state, the cluster state of the measurement-based computation model, and the history state used in the adiabatic simulation of a quantum circuit. We also show that the method, when adapted for quantum search, provides quadratic speedup as other optimal methods do with the advantages that the problem Hamiltonian is time independent and that the energy gap above the ground state is strictly nondecreasing with time. Likewise the method can be used for optimization as an alternative to the standard adiabatic algorithm.

Figure 1

(a) Interactions that appear in the toric code Hamiltonian. (b) Interactions in the cluster state Hamiltonian, following the same notation.

Figure 2

(Color online) Comparison of energy gap between stepwise adiabatic rotation to linear interpolation in the generation of the history state with $L=6$. For the stepwise adiabatic rotation, the gap never crosses the line corresponding to the final gap.