We 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.

翻译：我们提出一个针对有界延展量（相对于某结构化态类）量子态层析成像的通用框架。设 $\textsf{C}$ 为 $n$ 量子比特态族，满足：$(i)$ $\textsf{C}$ 可简洁表示，$(ii)$ 存在针对 $\textsf{C}$ 的弱不可知学习器。对于承诺可分解为 $|ψ\rangle = \sum_i c_i |φ_i\rangle$ （其中 $|φ_i\rangle \in \textsf{C}$ 且系数 $\ell_1$ 范数有界，我们称之为延展量）的未知态 $|ψ\rangle$，我们给出一个层析成像协议。主要贡献在于证明：针对 $\textsf{C}$ 的弱不可知学习器可提升为针对 $\textsf{C}$ 有界延展量态的层析算法。该归约是黑盒的，广泛适用于各类模型。作为应用，当 $\textsf{C}$ 为稳定子态类时，我们得到针对稳定子延展量 $\xi$ 至迹距离 $\varepsilon$ 的层析算法，时间复杂度为 $\textsf{poly}(n,(\xi/\varepsilon)^{\log(\xi/\varepsilon)})$；若假设高倍增域中的算法多项式Freiman-Ruzsa猜想，可改进为 $\textsf{poly}(n,\xi,1/\varepsilon)$。当未知态 $|ψ\rangle$ 为任意态时，我们给出一个算法分解结果——该结果在精神上类似于针对 $\textsf{C}$ 的量子态弱正则引理，并证明 $|ψ\rangle$ 中可由 $\textsf{C}$ 解释的结构可被高效学习。我们的核心概念启示是：对结构化基类的不可知学习可自动推出其低复杂度线性张成的可学习性。