Experimental measurement and a physical interpretation of quantum shadow enumerators

TL;DR

This paper links quantum shadow enumerators to Bell sampling experiments, enabling direct measurement of quantum weight enumerators and advancing understanding of entanglement in quantum error correction through theoretical and experimental insights.

Contribution

It establishes a rigorous framework for measuring quantum weight enumerators via Bell sampling, bridging quantum error correction and entanglement analysis with experimental validation.

Findings

Quantum shadow enumerators correspond to Bell sampling probabilities.

Framework for direct measurement of quantum weight enumerators is developed.

Experimental demonstration on a trapped-ion quantum computer confirms theoretical predictions.

Abstract

Throughout its history, the theory of quantum error correction has heavily benefited from translating classical concepts into the quantum setting. In particular, classical notions of weight enumerators, which relate to the performance of an error-correcting code, and MacWilliams' identity, which helps to compute enumerators, have been generalized to the quantum case. In this work, we establish a distinct relationship between the theoretical machinery of quantum weight enumerators and a seemingly unrelated physics experiment: we prove that Rains' quantum shadow enumerators - a powerful mathematical tool - arise as probabilities of observing fixed numbers of triplets in a Bell sampling experiment. This insight allows us to develop here a rigorous framework for the direct measurement of quantum weight enumerators, thus enabling experimental and theoretical studies of the entanglement structure of any quantum error-correcting code or state under investigation. On top of that, we derive concrete sample complexity bounds and physically-motivated robustness guarantees against unavoidable experimental imperfections. Finally, we experimentally demonstrate the possibility of directly measuring weight enumerators on a trapped-ion quantum computer. Our experimental findings are in good agreement with theoretical predictions and illuminate how entanglement theory and quantum error correction can cross-fertilize each other once Bell sampling experiments are combined with the theoretical machinery of quantum weight enumerators.