Continuous-variable state moments from randomized homodyne and heterodyne measurements

Abstract

Continuous-variable (CV) quantum states are naturally characterized by their moments, defined as expectation values of products of single- or multimode ladder operators. Many CV Hamiltonians and quantum algorithms are formulated directly in terms of these moments, and therefore an efficient procedure to estimate moments with limited state measurements is necessary. In this paper, we present a protocol for shadow tomography of moment-generating functions (MGFs) of CV states based on randomized homodyne and heterodyne measurements. The resulting shadows enable an efficient and concurrent estimation of many multimode moments. Our complexity analysis shows that the number of measurements required to estimate a certain moment to a given precision grows exponentially in the order of the moments. Finally, we assess the precision of these moment estimators by applying them to two tasks: detecting entanglement in both Gaussian and non-Gaussian states via the Shchukin-Vogel protocol, and characterizing optical loss in a photonic chip. We demonstrate that both tasks can be accomplished with only a few thousand randomized measurements. The small number of required measurements, combined with efficient sample processing, makes our protocol applicable to a wide range of CV simulation tasks.