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

TL;DR

This paper links quantum weight enumerators to the $n$-qubit parallelized SWAP test, enabling efficient estimation of quantum code distances and entanglement properties on quantum computers.

Contribution

It establishes a connection between quantum weight enumerators and the SWAP test, providing a physical interpretation and practical methods for their computation.

Findings

Shadow enumerators correspond to probabilities from the SWAP test.

The non-negativity of probabilities proves shadow inequalities.

Quantum code distances and entanglement measures can be efficiently estimated.

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.