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

• URL: http://arxiv.org/abs/2406.18280v1
• Date: Wed, 26 Jun 2024 12:06:40 GMT
• Title: Exploring quantum weight enumerators from the $n$-qubit parallelized SWAP test
• Authors: Fei Shi, Kaiyi Guo, Xiande Zhang, Qi Zhao,
• Abstract summary: We build the connection between quantum weight enumerators and the $n$-qubit parallelized SWAP test. We find that each shadow enumerator corresponds precisely to a probability in the $n$-qubit parallelized SWAP test. For applications, we employ the $n$-qubit parallelized SWAP test to determine the distances of quantum error-correcting codes.
• Score: 19.183393329155567
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum weight enumerators play a crucial role in quantum error-correcting codes and multipartite entanglement. They can be used to investigate the existence of quantum error-correcting codes and $k$-uniform states. In this work, we build the connection between quantum weight enumerators and the $n$-qubit parallelized SWAP test. We discover that each shadow enumerator corresponds precisely to a probability in the $n$-qubit parallelized SWAP test, providing a computable and operational meaning for the shadow enumerators. Due to the non-negativity of probabilities, we obtain an elegant proof for the shadow inequalities. Concurrently, we can also calculate the Shor-Laflamme enumerators and the Rains unitary enumerators from the $n$-qubit parallelized SWAP test. For applications, we employ the $n$-qubit parallelized SWAP test to determine the distances of quantum error-correcting codes, and the $k$-uniformity of pure states. Our results indicate that quantum weight enumerators can be efficiently estimated on quantum computers, and opening a path to calculate the distances of quantum error-correcting codes.

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