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arXiv · 2406.18280

Exploring quantum weight enumerators from the $n$-qubit parallelized SWAP test

Fei Shi·Kaiyi Guo·Xiande Zhang·Qi Zhao
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Abstract

Quantum weight enumerators are fundamental tools for analyzing quantum error-correcting codes and multipartite entanglement, offering insights into the existence of quantum error-correcting codes and $k$-uniform states. In this work, we establish a connection between quantum weight enumerators and the $n$-qubit parallelized SWAP test. We demonstrate that each shadow enumerator corresponds to a probability derived from this test, providing a physical interpretation for the shadow enumerators. Leveraging the non-negativity of these probabilities, we present an elegant proof for the shadow inequalities. Additionally, we show that the Shor-Laflamme weight enumerators and the Rains unitary enumerators can be calculated using the $n$-qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine the $k$-uniformity of pure states, and evaluate multipartite entanglement measures. Our results indicate that quantum weight enumerators can be efficiently estimated on quantum computers, opening a path to calculate and verify the distances of quantum error-correcting codes.

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Fei Shi, Kaiyi Guo, Xiande Zhang, Qi Zhao. 2024-06-26. Exploring quantum weight enumerators from the $n$-qubit parallelized SWAP test. https://arxiv.org/abs/2406.18280

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