{
  "$type": "site.standard.document",
  "bskyPostRef": {
    "cid": "bafyreidjjnyd4yx5oneimea6xqnlakh3dxsyeipy6xdy2chkooxtlc6qlm",
    "uri": "at://did:plc:3fychdutjjusoqeq24ljch6q/app.bsky.feed.post/3mntiqqxhrkb2"
  },
  "coverImage": {
    "$type": "blob",
    "ref": {
      "$link": "bafkreiflo6xt7is6b2iafwghkjahlgggocme5jwjsbeuqqwcywuvjhmszm"
    },
    "mimeType": "image/png",
    "size": 24783
  },
  "path": "/abs/2606.09728v1",
  "publishedAt": "2026-06-09T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Arpon Basu",
    "Joshua Brakensiek",
    "Pravesh K. Kothari",
    "Aaron Putterman"
  ],
  "textContent": "**Authors:** Arpon Basu, Joshua Brakensiek, Pravesh K. Kothari, Aaron Putterman\n\nIn this paper, we continue a line of research initiated by Basu, Brakensiek, and Putterman [2026] studying the sparsifiability of Hamiltonians. We focus particularly on the sparsifiability of the widely-studied Quantum Cut (QC) Hamiltonians. Our main result is that in an $n$-qubit system, any $n$-qubit QC Hamiltonian can be sparsified to $\\widetilde{O}(n /\\varepsilon^2)$ many terms while preserving the energy of every state up to a factor of $1 \\pm \\varepsilon$. Our result can be interpreted as giving an importance sampling scheme for the edges of an arbitrary graph $G$ such that the \\emph{Kikuchi} graph at level $\\ell$ of the sampled graph is a spectral approximation to the Kikuchi graph of $G$. Importantly, the \\emph{same} sampling scheme works simultaneously for all $\\ell$. The natural approach of leverage score sampling, analyzed via matrix concentration inequalities, yields a polynomially worse bound in our setting because the underlying matrices have dimension $\\sim 2^n$. Instead, our approach relies on decomposing the action of these matrices into invariant subspaces. Then, by using an operator-valued inequality of Alon and Kozma [Ann. Henri Poincaré, 2020], itself building on an \\emph{octopus inequality} of Caputo, Liggett, and Richthammer [J. AMS, 2010], we extend our sparsification technique to all expander graphs. We then invoke expander decomposition to extend our sparsifier to all graphs.",
  "title": "Quantum Cut Sparsifiers"
}