Fermi-Dirac machines as quantizations of neurons

2026-05-23

AI Summary

This work proposes a regularized quantization-based approach to construct quantum neurons by rigorously mapping classical activation functions to quantum observables acting on parameterized Hamiltonians. In the commuting case, the method exactly recovers classical neurons and, for the first time, provides quantum counterparts of widely used activations such as ReLU and GeLU, whose associated decision problems are proven to be BQP-complete. A trainable hybrid quantum-classical algorithm is devised by integrating Hamiltonian simulation, Hadamard tests, and continuous-variable quantization. Numerical experiments demonstrate that the proposed quantum Hamiltonian neurons can learn functions beyond the representational capacity of classical neurons, exhibiting a provable computational advantage that cannot be efficiently simulated classically.

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.

Problem

• quantum neuron
• canonical quantization
• Fermi-Dirac machine
• activation function
• BQP-completeness

Innovation

• canonical quantization
• quantum neuron
• activation observable
• hybrid quantum-classical algorithm
• BQP-completeness