Experimental Hybrid Shadow Tomography and Distillation

TL;DR

This paper introduces hybrid shadow tomography with a novel Fredkin gate to efficiently estimate nonlinear quantum functions, reducing measurement costs and enabling high-purity state extraction in quantum systems.

Contribution

The work develops and experimentally demonstrates a hybrid shadow protocol using a deterministic Fredkin gate, significantly improving nonlinear quantum state estimation and virtual distillation techniques.

Findings

High process fidelity of 0.935 for the Fredkin gate.

Reduced sample complexity in nonlinear function estimation.

Successful implementation of virtual distillation for high-purity states.

Abstract

Characterization of quantum states is a fundamental requirement in quantum science and technology. As a promising framework, shadow tomography shows significant efficiency in estimating linear functions, however, for the challenging nonlinear ones, it requires measurements at an exponential cost. Here, we implement an advanced shadow protocol, so-called hybrid shadow~(HS) tomography, to reduce the measurement cost in the estimation of nonlinear functions in an optical system. We design and realize a deterministic quantum Fredkin gate with single photon, achieving high process fidelity of $0.935\pm0.001$. Utilizing this novel Fredkin gate, we demonstrate HS in the estimations, like the higher-order moments up to 4, and reveal that the sample complexity of HS is significantly reduced compared with the original shadow protocol. Furthermore, we utilize these higher-degree functions to implement virtual distillation, which effectively extracts a high-purity quantum state from two noisy copies. The virtual distillation is also verified in a proof-of-principle demonstration of quantum metrology, further enhancing the accuracy of parameter estimation. Our results suggest that HS is efficient in state characterization and promising for quantum technologies.