Learning quantum graph states with product measurements

TL;DR

This paper presents an explicit algorithm for learning quantum graph states using product measurements, achieving high accuracy with fewer copies than theoretical lower bounds, and analyzes error bounds under depolarizing noise.

Contribution

It introduces a novel explicit algorithm for learning quantum graph states with product measurements and provides bounds on the number of copies needed, including under noise conditions.

Findings

Algorithm successfully learns graph states with high probability.

Number of copies needed scales with graph degree and desired accuracy.

Bounds established for learning under depolarizing errors.

Abstract

We consider the problem of learning $N$ identical copies of an unknown $n$-qubit quantum graph state with product measurements. These graph states have corresponding graphs where every vertex has exactly $d$ neighboring vertices. Here, we detail an explicit algorithm that uses product measurements on multiple identical copies of such graph states to learn them. When $n \gg d$ and $N = O(d \log(1/\epsilon) + d^2 \log n ),$ this algorithm correctly learns the graph state with probability at least $1- \epsilon$. From channel coding theory, we find that for arbitrary joint measurements on graph states, any learning algorithm achieving this accuracy requires at least $\Omega(\log (1/\epsilon) + d \log n)$ copies when $d=o(\sqrt n)$. We also supply bounds on $N$ when every graph state encounters identical and independent depolarizing errors on each qubit.