  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 G. 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/G^2, and a method based on fixed-point search, has a maximum cost of order
1/G^(3/2). Additionally, if the Hamiltonians satisfy a frustration-free
property, such costs can be further improved to order 1/G^(3/2) and 1/G,
respectively. Our methods offer an important advantage over adiabatic quantum
computation when the gap is small, where the cost is of order 1/G^3.
