Exploring Quantum Weight Enumerators From the n-Qubit Parallelized SWAP Test
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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 <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-uniform states. In this work, we establish a connection between quantum weight enumerators and the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-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.