Experimental quantum compressed sensing for a seven-qubit system

TLDR

Quantum devices are limited by the exponential scaling of quantum tomography, but quantum compressed sensing can reconstruct states from incomplete measurements, a technique originally proposed for low‑rank density matrices. This study experimentally implements compressed tomography on a seven‑qubit trapped‑ion system, the largest such realization, and introduces numerical methods to scale reconstruction to this dimension while arguing that low‑rank estimates remain valid in general. The authors reconstruct the state of a seven‑qubit topological color code using only 127 Pauli measurement settings repeated 100 times each, applying their new numerical reconstruction algorithms. They find that statistical noise limits reliable reconstruction to the leading eigenvectors, while the remaining eigenvectors follow a random‑matrix distribution that carries no information about the true state.

Abstract

Well-controlled quantum devices with their increasing system size face a new roadblock hindering further development of quantum technologies: The effort of quantum tomography---the characterization of processes and states within a quantum device---scales unfavorably to the point that state-of-the-art systems can no longer be treated. Quantum compressed sensing mitigates this problem by reconstructing the state from an incomplete set of observables. In this work, we present an experimental implementation of compressed tomography of a seven qubit system---the largest-scale realization to date---and we introduce new numerical methods in order to scale the reconstruction to this dimension. Originally, compressed sensing has been advocated for density matrices with few non-zero eigenvalues. Here, we argue that the low-rank estimates provided by compressed sensing can be appropriate even in the general case. The reason is that statistical noise often allows only for the leading eigenvectors to be reliably reconstructed: We find that the remaining eigenvectors behave in a way consistent with a random matrix model that carries no information about the true state. We report a reconstruction of quantum states from a topological color code of seven qubits, prepared in a trapped ion architecture, based on tomographically incomplete data involving 127 Pauli basis measurement settings only, repeated 100 times each.