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 |Φ0(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=∫01 |∂s |Φ0(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∈[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 γ1/p. A typical avoided crossing is in width class p=1, which gives L=O(√(log Δ*−1)) and K=O(log 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.
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BibTeX
@misc{han2026bounds,
title = {{Bounds on adiabatic path geometry from the width class of the gap profile}},
author = {Mancheon Han and Sangkook Choi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39286},
eprint = {2609.39286},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39286},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounds on adiabatic path geometry from the width class of the gap profile
AU - Mancheon Han
AU - Sangkook Choi
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.39286
UR - https://arxiv.org/abs/2609.39286
AB - 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=∫_0^1 |∂_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∈[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(√(log Δ_*^−1)) and K=O(log 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.
ER -