Quantum state tomography via compressed sensing

TL;DR

This paper introduces compressed sensing techniques for efficient quantum state tomography, significantly reducing measurement requirements for nearly pure states and demonstrating practical advantages through theoretical analysis and simulations.

Contribution

It develops compressed sensing methods tailored for quantum states, enabling accurate reconstruction with fewer measurements and practical implementation features.

Findings

Reconstructs quantum states with O(rd log^2 d) measurements

Methods are stable against noise and applicable to approximately low-rank states

Numerical simulations confirm theoretical performance bounds

Abstract

We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rd log^2 d) measurement settings, compared to standard methods that require d^2 settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low-rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed. We present both theoretical bounds and numerical simulations.