Bounding many-body properties under partial information and finite measurement statistics

TL;DR

This paper develops scalable methods using moment-matrix relaxations and semidefinite programming to bound properties of many-body quantum systems from incomplete and noisy measurement data, aiding quantum system analysis.

Contribution

It introduces a scalable formalism that incorporates system-specific knowledge and shot noise, improving bounds on many-body quantum properties from finite data.

Findings

Scalable bounds derived from moment-matrix relaxations.

Inclusion of system-specific information enhances bounds.

Method effectively handles shot noise in experimental data.

Abstract

Calculating bounds of properties of many-body quantum systems is of paramount importance, since they guide our understanding of emergent quantum phenomena and complement the insights obtained from estimation methods. Recent semidefinite programming approaches enable probabilistic bounds from finite-shot measurements of easily accessible, yet informationally incomplete, observables. Here we render these methods scalable in the number of qubits by instead utilizing moment-matrix relaxations. After introducing the general formalism, we show how the approach can be adapted with specific knowledge of the system, such as it being the ground state of a given Hamiltonian, possessing specific symmetries or being the steady state of a given Lindbladian. Our approach defines a scalable real-world certification scheme leveraging semidefinite programming relaxations and experimental estimations which, unavoidably, contain shot noise.