Fermi-Dirac machines as quantizations of neurons
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Quantum Physics arXiv:2605.24386 (quant-ph) [Submitted on 23 May 2026] Title:Fermi-Dirac machines as quantizations of neurons Authors:Alexander He, Nana Liu, Mark M. Wilde View a PDF of the paper titled Fermi-Dirac machines as quantizations of neurons, by Alexander He and 2 other authors View PDF HTML (experimental) Abstract:Fermi-Dirac machines were proposed recently as an approach to solving semidefinite optimization problems on quantum computers. Here, we reinterpret them as canonical quantizations of classical neurons. By viewing a classical neuron as an activation function applied to a parameterized classical Hamiltonian, we quantize this model by replacing classical variables with operators whose eigenvalues encode their possible values. This follows the standard approach to canonical quantization in quantum mechanics. Crucially, when the Hamiltonian consists of commuting operators, our construction reduces exactly to a classical neuron. More generally, our approach yields an activation observable, defined as an activation function applied to a parameterized quantum Hamiltonian. The output of this quantized neuron is a random variable with expectation value equal to that of the activation observable with respect to an input state. We develop efficient hybrid quantum-classical algorithms for evaluating outputs and gradients of our quantized neurons, enabling evaluation and training. These algorithms rely on basic primitives that include random sampling, Hamiltonian simulation, and the Hadamard test. We also quantize a whole host of other activation functions, including the smooth rectified linear unit (ReLU), sigmoid linear unit, Gaussian-smoothed ReLU, and Gaussian error linear unit (GeLU), which are known to be useful for deep learning applications. Numerical experiments indicate that neurons based on quantum Hamiltonians can learn functions that classical neurons cannot. We further define a computational decision problem based on Fermi-Dirac neurons and prove that it is BQP-complete, providing complexity-theoretic evidence against efficient classical simulation. Finally, we generalize our approach to continuous quantum variables and sketch two different ways of composing these neurons into networks. Comments: Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG) Cite as: arXiv:2605.24386 [quant-ph] (or arXiv:2605.24386v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.24386 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Mark Wilde [view email] [v1] Sat, 23 May 2026 04:09:03 UTC (847 KB) Full-text links: Access Paper: View a PDF of the paper titled Fermi-Dirac machines as quantizations of neurons, by Alexander He and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: cond-mat cond-mat.stat-mech cs cs.DS cs.LG References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)