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 $\mathrm{\mid }{\mathrm{\Phi }}_0(s)⟩$ 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 $L={\int }_0^1\mathrm{\parallel }{\mathrm{\partial }}_s\mathrm{\mid }{\mathrm{\Phi }}_0(s)⟩\mathrm{\parallel }ds$ and its total curvature $K$. This geometry affects the evolution time $T$. For instance, $T$ is at least of order $L\mathrm{/}{\mathrm{\Delta }}_{\ast }$, where ${\mathrm{\Delta }}_{\ast }$ is the minimum energy gap over $s\in [0,1]$. The traditional approach gives $L=\mathcal{O}({\mathrm{\Delta }}_{\ast }^{-1\mathrm{/}2})$ and $K=\mathcal{O}({\mathrm{\Delta }}_{\ast }^{-1})$, which are loose in many cases. We improve these bounds by using ${\mu }_{<}(\gamma )$, the width of the region in $s$ on which the gap is below $\gamma$. The gap profile is in width class $p$ when ${\mu }_{<}(\gamma )$ falls at least as fast as ${\gamma }^{1\mathrm{/}p}$. A typical avoided crossing is in width class $p=1$, which gives $L=\mathcal{O}(\sqrt{\log {\mathrm{\Delta }}_{\ast }^{-1}})$ and $K=\mathcal{O}(\log {\mathrm{\Delta }}_{\ast }^{-1})$. A profile in width class $p>1$ gives power laws with exponents $(p-1)\mathrm{/}(2p)$ for $L$ and $(p-1)\mathrm{/}p$ for $K$. The width class also bounds the evolution time. At $p=1$ we have $T=\mathcal{O}({\mathrm{\Delta }}_{\ast }^{-2})$ with a schedule that advances $s$ at a constant rate, and $T=\mathcal{O}({\mathrm{\Delta }}_{\ast }^{-1})$ 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 ${\mathrm{\Delta }}_{\ast }$ for every integer $p\ge 1$, 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.