Bounds on adiabatic path geometry from the width class of the gap profile

In an adiabatic evolution, the wave function follows the path of ground states Φ 0(s) of a Hamiltonian H(s) as s is tuned from 0 to 1. The geometry of this adiabatic path is described by quantities such as its length L= integral 0 1 | d s Φ 0(s)|,ds and its total curvature K. This geometry affects the evolution time T. For instance, T is at least of order L/Δ*, where Δ* is the minimum energy gap over s in[0,1]. The traditional approach gives L=O(Δ*-1/2) and K=O(Δ*-1), which are loose in many cases. We improve these bounds by using μ<(γ), the width of the region in s on which the gap is below γ. The gap profile is in width class p when μ<(γ) falls at least as fast as γ 1/p. A typical avoided crossing is in width class p=1, which gives L=O(sqrt Δ*-1) and K=O(Δ*-1). A profile in width class p>1 gives power laws with exponents (p-1)/(2p) for L and (p-1)/p for K. The width class also bounds the evolution time. At p=1 we have T=O(Δ*-2) with a schedule that advances s at a constant rate, and T=O(Δ*-1) up to a polylogarithmic correction with a schedule that traverses the path at a constant geometric speed. The bounds on L hold for any twice continuously differentiable H(s), and the bounds on K for affine H(s). We also prove the tightness of scaling of L in Δ* for every integer p 1, and the scaling of K at p=1. We demonstrate the bounds on the adiabatic Grover search, the XXZ spin chain, and molecular electronic Hamiltonians.