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  "path": "/abs/2606.07425v1",
  "publishedAt": "2026-06-08T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Srinivasan Arunachalam",
    "Arkopal Dutt"
  ],
  "textContent": "**Authors:** Srinivasan Arunachalam, Arkopal Dutt\n\nWe give a general framework for tomography of states that have bounded-extent with respect to a structured class of states. Let $\\textsf{C}$ be a family of $n$-qubit states such that: $(i)$ $\\textsf{C}$ is succinctly representable and $(ii)$ there is a weak agnostic learner of $\\textsf{C}$. We give a tomography protocol for an unknown state $|ψ\\rangle$ that is promised to admit a decomposition of the form $|ψ\\rangle = \\sum_i c_i |φ_i\\rangle$, where $|φ_i\\rangle \\in \\textsf{C}$ with bounded $\\ell_1$-norm of the coefficients (which we call extent). Our main contribution is to show that a weak agnostic learner for $\\textsf{C}$ can be boosted into a tomography algorithm for states with bounded extent with respect to $\\textsf{C}$. Our reduction is black-box and applies broadly across model classes. As an application, when $\\textsf{C}$ is the class of stabilizer states, we obtain tomography algorithms for states with stabilizer extent $ξ$ up to trace distance $\\varepsilon$, in time $\\textsf{poly}(n,(ξ/\\varepsilon)^{\\log(ξ/\\varepsilon)})$, which is improvable to $ \\textsf{poly}(n,ξ,1/\\varepsilon)$ assuming the algorithmic polynomial Freiman-Ruzsa conjecture in the high-doubling regime. When the unknown state $|ψ\\rangle$ is arbitrary, we give an algorithmic decomposition result in the spirit of a weak regularity lemma for quantum states with respect to $\\textsf{C}$ and show that the structure in $|ψ\\rangle$ that is explainable by $\\textsf{C}$ can be efficiently learned. Our main conceptual message is that agnostic learning of a structured base class automatically yields learnability of its low-complexity linear span.",
  "title": "Tomography of quantum states with bounded extent"
}