Improved bounds for eigenpath traversal

We present a bound on the length of the path defined by the ground states of a continuous family of Hamiltonians in terms of the spectral gap Δ. We use this bound to obtain a significant improvement over the cost of recently proposed methods for quantum adiabatic state transformations and eigenpath traversal. In particular, we prove that a method based on evolution randomization, which is a simple extension of adiabatic quantum computation, has an average cost of order 1/Δ², and a method based on fixed-point search has a maximum cost of order 1/Δ3/2. Additionally, if the Hamiltonians satisfy a frustration-free property, such costs can be further improved to order 1/Δ3/2 and 1/Δ, respectively. Our methods offer an important advantage over adiabatic quantum computation when the gap is small, where the cost is of order 1/Δ³.