# Fermi-Dirac machines as quantizations of neurons

arXiv: https://arxiv.org/abs/2605.24386 · HTML: https://mlsanity.com/paper/2605.24386/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 100 references that are in the map.
- First posted on arXiv 23 May 2026.

## Builds on (most cited first, 13 of 13)

- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Searching for Activation Functions](https://mlsanity.com/paper/1710.05941/) (2017)
- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Quantum Neuron: an elementary building block for machine learning on quantum computers](https://mlsanity.com/paper/1711.11240/) (2017)
- [Quantum Perceptron Models](https://mlsanity.com/paper/1602.04799/) (2016)
- [Activation Functions in Artificial Neural Networks: A Systematic Overview](https://mlsanity.com/paper/2101.09957/) (2021)
- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)
- [Fermi-Dirac thermal measurements: A framework for quantum hypothesis testing and semidefinite optimization](https://mlsanity.com/paper/2603.04061/) (2026)
- [Fast randomized entropically regularized semidefinite programming](https://mlsanity.com/paper/2303.12133/) (2023)
- [Natural Quantization of Neural Networks](https://mlsanity.com/paper/2503.15482/) (2025)
- [Quantum Fisher information matrices from R'enyi relative entropies](https://mlsanity.com/paper/2510.02218/) (2025)

## Built on by (most cited first, 3 of 3)

- [Canonical quantization of neurons](https://mlsanity.com/paper/2607.05000/) (2026)
- [Deep Learning for Semen Analysis in Male Infertility: Computer Vision, Multimodal Fusion, and Clinical Translation](https://mlsanity.com/paper/2607.05311/) (2026)
- [Unifying quantum measurement constructions via a relative-entropy minimum change principle](https://mlsanity.com/paper/2608.04055/) (2026)

## Abstract (arXiv)

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.