Canonical quantization of neurons

2026-07-06

AI Summary

This work proposes a systematic approach to transform classical neurons into fundamental building blocks for quantum machine learning, enabling the learning of unknown observables from labeled quantum data. The method models a neuron as a composition of an energy function and an activation function, mapping the energy component to a quantum Hamiltonian via canonical quantization and quantizing the activation function through matrix functional calculus to yield a measurable activation observable. A hybrid quantum-classical training algorithm is then constructed by integrating Hadamard tests, Hamiltonian simulation, and single-copy power methods. As the first framework to systematically apply canonical quantization to neuron modeling, it offers strong theoretical grounding and scalability. Numerical experiments demonstrate that the resulting quantum neuron exhibits superior expressivity over its classical counterpart on representative tasks, highlighting its potential for quantum function approximation.

Abstract

Canonical quantization provides a systematic procedure for constructing quantum models from classical Hamiltonians. Here, we apply this principle to a fundamental computational primitive of machine learning: the neuron. Specifically, by viewing a neuron as a composition of an energy function and an activation function, we quantize this model by replacing the energy function with a quantum Hamiltonian and applying the activation function to it through matrix functional calculus. This results in an activation observable that can be measured on an input quantum state. We investigate the use of these quantized neurons for function approximation, where the objective is to learn an unknown observable from labeled quantum data. For this purpose, we develop hybrid quantum-classical algorithms for training and evaluation, including procedures for measuring the activation observable and estimating gradients of the squared loss error. Our algorithms for gradient estimation rely on basic primitives like classical random sampling, the Hadamard test, and Hamiltonian simulation, and those for measuring an activation observable rely on quantum algorithms known as the power of one qumode and Schroedingerization. Numerical experiments demonstrate that our quantized neurons exhibit enhanced expressive capabilities relative to corresponding classical neurons on representative learning tasks. Our work establishes canonical quantization as a principled framework for constructing quantum machine learning primitives and provides a foundation for developing neural architectures tailored to quantum data.

Problem

• canonical quantization
• quantum neuron
• function approximation
• quantum machine learning
• quantum observable

Innovation

• canonical quantization
• quantum neuron
• activation observable
• hybrid quantum-classical algorithm
• function approximation