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Bounds on adiabatic path geometry from the width class of the gap profile — arXiv:2609.39286

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.39286
License (as archived): CC0-1.0
Archived: 2026-10-07T15:21:16.956Z
SHA-256 of the archived text: 5e95aadd6299da2527feb92c7ca6462e8bea826fceace3a0fe22dc2c69568b18
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How to cite

Bounds on adiabatic path geometry from the width class of the gap profile — arXiv:2609.39286 — arXiv (en), https://arxiv.org/abs/2609.39286 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bb5803e802ea (Metraipolis archive dc_bb5803e802ea, sha256 5e95aadd6299da2527feb92c7ca6462e8bea826fceace3a0fe22dc2c69568b18)

BibTeX

@misc{dcbb5803e802ea,
  title = {Bounds on adiabatic path geometry from the width class of the gap profile — arXiv:2609.39286},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.39286},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bb5803e802ea; sha256 5e95aadd6299da2527feb92c7ca6462e8bea826fceace3a0fe22dc2c69568b18},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.39286. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $L=O(\Delta_*^{-1/2})$ and $K=O(\Delta_*^{-1})$, 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 $L=O(\sqrt{\log \Delta_*^{-1}})$ and $K=O(\log \Delta_*^{-1})$. 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 $T=O(\Delta_*^{-2})$ with a schedule that advances $s$ at a constant rate, and $T=O(\Delta_*^{-1})$ 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.

Tema degli esploratori: fisica e astronomia (retro-catalogo arXiv: annunci del 2026-10-01 in quant-ph).
Categorie arXiv: Quantum Physics (quant-ph).
Pagina del lavoro: https://arxiv.org/abs/2609.39286 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T14:53:56.673Z dalla pagina https://arxiv.org/catchup/quant-ph/2026-10-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).