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On the Entanglement and Mixedness of Quantum Boolean Function Circuits

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Machines, Computations, and Universality (MCU 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15270))

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Abstract

Understanding the entanglement and mixedness introduced into a quantum system by quantum Boolean functions (BFs) circuits holds a significant potential in quantum information theory. This study provides an overview and empirical findings on purity, negativity, and von Neumann entropy with the aim to reveal an insight into the internal structure of the corresponding unitary operators. Approximately 5,000 quantum circuits were examined for BFs of different types, e.g., balanced, symmetric, bent, etc. and of various properties such as algebraic degree, algebraic immunity, resiliency order, and others. While previous research typically focuses on input superposition alone through addition of Hadamard gates (Superposition scenario), we introduce additional entanglement through random statevector generation (Random Statevector scenarios). Results show a slight increase in von Neumann entropy and negativity for the Random Statevector scenario as compared to the Superposition scenario. Surprisingly, further measurement increases were observed when adding classical randomness via random qubit rotations (Random Qubit Rotations scenario) instead of random statevector generation. The observed trends include lower mean values and wider ranges of negativity and von Neumann entropy for BFs of higher algebraic degree in the Superposition scenario. Additionally, plateaued functions of lower algebraic degree exhibit purity measurements with wider spread in the Random Statevector scenario as compared to the Superposition scenario. We hope that this work may inspire new theoretical studies on quantum BF circuits. For example, a deeper understanding of quantum BF circuits with high mixedness and entanglement may lead to the development of novel quantum error correction techniques.

This study was supported by the IBM Quantum Researchers program.

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Acknowledgements

The authors appreciate the valuable resources provided through the IBM Quantum Researcher’s program.

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Authors and Affiliations

  1. School of Computing, University of North Florida, Jacksonville, FL, 32224, USA

    Zornitza Prodanoff, Iliya Kulbaka & Natasha Interlichia

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  1. Zornitza Prodanoff
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  2. Iliya Kulbaka
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  3. Natasha Interlichia
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Correspondence to Zornitza Prodanoff .

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Editors and Affiliations

  1. Université Côte d'Azur, Sophia Antipolis, France

    Enrico Formenti

  2. University of Orléans, Orléans, France

    Jérôme Durand-Lose

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Prodanoff, Z., Kulbaka, I., Interlichia, N. (2025). On the Entanglement and Mixedness of Quantum Boolean Function Circuits. In: Formenti, E., Durand-Lose, J. (eds) Machines, Computations, and Universality. MCU 2024. Lecture Notes in Computer Science, vol 15270. Springer, Cham. https://doi.org/10.1007/978-3-031-81202-6_9

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  • DOI: https://doi.org/10.1007/978-3-031-81202-6_9

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