Tomography of quantum states with bounded extent

arXiv:2606.07425v1 Announce Type: cross Abstract: 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 $|\psi\rangle$ that is promised to admit a decomposition of the form $|\psi\rangle = \sum_i c_i |\phi_i\rangle$, where $|\phi_i\rangle \in \textsf{C}$ 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 $\xi$ up to trace distance $\varepsilon$, in time $\textsf{poly}(n,(\xi/\varepsilon)^{\log(\xi/\varepsilon)})$, which is improvable to $ \textsf{poly}(n,\xi,1/\varepsilon)$ assuming the algorithmic polynomial Freiman-Ruzsa conjecture in the high-doubling regime. When the unknown state $|\psi\rangle$ 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 $|\psi\rangle$ 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.