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arXiv:2406.18280

Exploring quantum weight enumerators from the nnn-qubit parallelized SWAP test

Fei Shi, Kaiyi Guo, Xiande Zhang, Qi Zhao

Published Jun 26, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2406.18280

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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 kkk-uniform states. In this work, we establish a connection between quantum weight enumerators and the nnn-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 nnn-qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine the kkk-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.
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 andk-uniform states. In this work, we establish a connection between quantum weight enumerators and then-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 then-qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine thek-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.
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 links enumerators of a code to the ones of its dual, have been generalized to the quantum case. In this work, we establish a 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 two-copy 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 quantum state under investigation. On top of that, we derive concrete sample complexity bounds and physically motivated robustness guarantees against unavoidable experimental imperfections. Finally, we demonstrate the feasibility of experimentally learning 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 cross-fertilize each other once two-copy Bell sampling experiments are combined with the theoretical machinery of quantum weight enumerators.
In a previous paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-02, 1997) define two "weight enumerators" for quantum error-correcting codes, connected by a MacWilliams (1977) transform, and use them to give a linear-programming bound for quantum codes. We extend their work by introducing another enumerator, based on the classical theory of shadow codes, that tightens their bounds significantly. In particular, nearly all of the codes known to be optimal among additive quantum codes (codes derived from orthogonal geometry) can be shown to be optimal among all quantum codes. We also use the shadow machinery to extend a bound on additive codes to general codes, obtaining as a consequence that any code of length, can correct at most [(n+1)/6] errors.
A unified Fourier framework for quantum weight enumerators of prime-power-dimensional qudits is developed in this paper. We introduce the notions of Weyl power spectrum and Weyl-conjugation autocorrelation for the Weyl-operator quantities nderlying the Shor--Laflamme weight enumerators. Representing these quantities as formal vectors, we show that they form a symplectic Fourier transform pair, from which the quantum MacWilliams identity follows by applying the Hamming-weight functional. This framework extends to a family of twisted Fourier transforms, including the Rains shadow transform. We show that identical-copy Bell sampling corresponds to a twisted autocorrelation related to the shadow enumerator through Weyl translation and Hamming projection. We further prove that all twisted Fourier transforms are Weyl translations of either the ordinary MacWilliams transform or the Rains shadow transform. This two-class reduction implies that the twists yield no additional independent coefficientwise positivity constraints under compatible projections. Nevertheless, the framework leads to refined MacWilliams and shadow identities that retain more Fourier information than ordinary Hamming weight enumerators. We employ these refined identities in linear programming, together with simplex mapping and shadow-coset counting, to exclude surviving candidate weight distributions. These techniques yield tighter bounds for binary and qutrit quantum codes, including nonexistence results for specific code parameters.
In a recent paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-2, 1997) defined two "weight enumerators" for quantum error-correcting codes, connected by a MacWilliams transform, and used them to give a linear programming bound for quantum codes. We introduce two new enumerators which, while much less powerful at producing bounds, are useful tools nonetheless. The new enumerators are connected by a much simpler duality transform, clarifying the duality between Shor and Laflamme's enumerators. We also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance, and to extend the enumerator theory to codes with block size greater than 2.
As emerging quantum architectures evolve into heterogeneous networks combining different physical substrates, such as qubits for logic and higher-dimensional qudits for robust communication, the traditional scalar metrics of quantum error correction become insufficient. To address this, we introduce a mathematical framework based on dimension multisets to characterize quantum error-correcting codes (QECC) and absolutely maximally entangled (AME) states in mixed-dimensional Hilbert spaces. By replacing scalar weights with multisets, we accurately capture the exact physical composition of error supports across these diverse systems. Our central result is the mixed-dimensional quantum MacWilliams identity, which establishes the formal algebraic relationship between Shor-Laflamme enumerators and unitary weight enumerators. From this foundation, we deduce the mixed-dimensional shadow identity and derive rigorous, generalized constraints on code parameters, explicitly formulating the mixed-dimensional quantum Hamming, Singleton and Scott bounds, and developing a linear program to systematically evaluate code viability. For the Singleton bound, a tighter bound that has no homogeneous analogue is derived for pure mixed-dimensional codes. Finally, we deploy this enumerator machinery to thoroughly analyze AME states, utilizing shadow inequalities to constrain their existence and introducing a combinatorial grid method for the explicit construction of mixed-dimensional tripartite AME states.