Tomography of quantum states with bounded extent

2026-06-05

AI Summary

This work addresses the problem of efficiently performing tomography on an unknown quantum state that can be expressed as a low-ℓ₁-norm linear combination of states from a structured class. We propose a general framework that, for the first time, black-box boosts a weak agnostic learner over such a class into an efficient tomography protocol for states of bounded extent, thereby revealing a deep connection between quantum learning and state tomography. Our approach integrates techniques from weak agnostic learning, quantum state decomposition, stabilizer structures, and an algorithmic regularity lemma. As an application, we achieve tomography within trace distance ε for states in the stabilizer class with time complexity poly(n, (ξ/ε)^{log(ξ/ε)}), which can be further improved under the polynomial Freiman–Ruzsa conjecture.

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 $|ψ\rangle$ 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.

Problem

• quantum state tomography
• bounded extent
• agnostic learning
• stabilizer states
• quantum complexity

Innovation

• quantum state tomography
• bounded extent
• agnostic learning
• stabilizer states
• black-box reduction