Optimal Overlapping Tomography

TL;DR

This paper develops optimal protocols for overlapping quantum tomography, significantly reducing measurement settings needed to efficiently characterize large quantum systems, with practical demonstrations and broad applications.

Contribution

It introduces graph theory-based algorithms for minimal measurement settings and proves optimality of projective measurements for reconstructing k-body marginals.

Findings

Reduced measurement settings for two-body tomography in planar qubit systems.

Proved that $3^k$ settings suffice for reconstructing all k-body marginals.

Experimental demonstration with six-photon quantum system.

Abstract

Characterising large-scale quantum systems is central to fundamental physics and essential for applications of quantum technologies. While a full characterisation requires exponentially increasing resources, focusing on application-relevant information can often lead to significantly simplified analysis. Overlapping tomography is such a scheme, allowing one to obtain all the information contained in specific subsystems of multiparticle quantum systems in an efficient manner, but the ultimate limits of this approach remain elusive. We present protocols for overlapping tomography that are optimal with respect to the number of measurement settings. First, by providing algorithmic approaches based on graph theory we find the minimal number of Pauli settings, relating overlapping tomography to the problem of covering arrays in combinatorics. This significantly reduces the number of measurement settings, showing for instance that two-body overlapping tomography of nearest neighbours in qubit systems with planar topologies can always be performed with nine Pauli settings. Second, we prove that using general projective measurements, all $k$-body marginals can be reconstructed with only $3^k$ settings, independently of the system size. Finally, we demonstrate the practical applicability of our methods in a six-photon experiment. Our results will find applications in learning noise and interaction patterns in quantum computers as well as characterising fermionic systems in quantum chemistry.