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

Abstract:In an adiabatic evolution, the wave function follows the path of ground states $|\Phi_0(s)\rangle$ 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 $L=\int_0^1 \|\partial_s |\Phi_0(s)\rangle\|\,ds$ and its total curvature $K$. This geometry affects the evolution time $T$. For instance, $T$ is at least of order $L/\Delta_*$, where $\Delta_*$ is the minimum energy gap over $s\in[0,1]$. The traditional approach gives $L=\mathcal{O}(\Delta_*^{-1/2})$ and $K=\mathcal{O}(\Delta_*^{-1})$, 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 $\gamma^{1/p}$. A typical avoided crossing is in width class $p=1$, which gives $L=\mathcal{O}(\sqrt{\log \Delta_*^{-1}})$ and $K=\mathcal{O}(\log \Delta_*^{-1})$. 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 $T=\mathcal{O}(\Delta_*^{-2})$ with a schedule that advances $s$ at a constant rate, and $T=\mathcal{O}(\Delta_*^{-1})$ 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 $\Delta_*$ for every integer $p\ge1$, 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.

arXiv:2609.39286 [quant-ph]

(or arXiv:2609.39286v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.39286 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.