Agnostic Tomography of Stabilizer Product States

TL;DR

This work defines agnostic tomography for quantum states and delivers the first efficient algorithm for learning stabilizer product states under this model. By leveraging Bell difference sampling and the structure of stabilizer products, the authors show that, when the unknown state has fidelity at least $\tau$ with a stabilizer product state, one can recover a near-optimal stabilizer-product description in time $n^{O(1+\log(1/\tau))}/\varepsilon^2$; the approach combines entropy-based sampling guarantees with a clique-based reconstruction of the stabilizer group. The result demonstrates robustness to noise and closes a gap between sample-efficient learning and computational efficiency for a nontrivial quantum state class. The work also situates agnostic tomography within a broader program of robust quantum learning and opens directions for extending efficient agnostic learners to additional state families and process-tomography tasks.

Abstract

We define a quantum learning task called agnostic tomography, where given copies of an arbitrary state $ρ$ and a class of quantum states $\mathcal{C}$, the goal is to output a succinct description of a state that approximates $ρ$ at least as well as any state in $\mathcal{C}$ (up to some small error $\varepsilon$). This task generalizes ordinary quantum tomography of states in $\mathcal{C}$ and is more challenging because the learning algorithm must be robust to perturbations of $ρ$. We give an efficient agnostic tomography algorithm for the class $\mathcal{C}$ of $n$-qubit stabilizer product states. Assuming $ρ$ has fidelity at least $τ$ with a stabilizer product state, the algorithm runs in time $n^{O(1 + \log(1/τ))} / \varepsilon^2$. This runtime is quasipolynomial in all parameters, and polynomial if $τ$ is a constant.

Theorems & Definitions (24)

• Definition 1.1: Agnostic tomography
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2: Based on Can20
• proof : Proof sketch
• Lemma 3.1
• proof
• Lemma 3.2
• proof