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Neural network representation of quantum systems

Koji Hashimoto, Yuji Hirono, Jun Maeda, Jojiro Totsuka-Yoshinaka

TL;DR

A novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters, which can be applied to interacting quantum systems / field theories, even away from the Gaussian limit.

Abstract

It has been proposed that random wide neural networks near Gaussian process are quantum field theories around Gaussian fixed points. In this paper, we provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. Our simple idea is to use the universal approximation theorem of neural networks to generate arbitrary paths in the Feynman's path integral. The map can be applied to interacting quantum systems / field theories, even away from the Gaussian limit. Our findings bring machine learning closer to the quantum world.

Neural network representation of quantum systems

TL;DR

A novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters, which can be applied to interacting quantum systems / field theories, even away from the Gaussian limit.

Abstract

It has been proposed that random wide neural networks near Gaussian process are quantum field theories around Gaussian fixed points. In this paper, we provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. Our simple idea is to use the universal approximation theorem of neural networks to generate arbitrary paths in the Feynman's path integral. The map can be applied to interacting quantum systems / field theories, even away from the Gaussian limit. Our findings bring machine learning closer to the quantum world.
Paper Structure (16 sections, 69 equations, 6 figures, 1 table)

This paper contains 16 sections, 69 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. A brief review of the Neural Network Field Theory
  3. Quantum mechanics as a neural network
  4. Path integral and statistical summation of network parameters
  5. Example 1 : Free particle
  6. Example 2 : Harmonic oscillator
  7. Example 3 : Quantum field theory
  8. Various activation functions and representation of quantum systems
  9. Step function activation
  10. Cosine activation and relation to the NNFT
  11. Quantum mechanics with Euclidean time as a neural network
  12. Ground state wave function
  13. Random walk interpretation and stochastic quantization
  14. Summary and discussions
  15. Generality of our network via gauge symmetry
  16. ...and 1 more sections

Figures (6)

  • Figure 1: Left: a path of $x(t)$ in the quantum-mechanical path-integral. Right: a multilayer perceptron, our neural network architecture.
  • Figure 2: Left: a multilayer perceptron representing a quantum mechanics of a particle in three dimensional space. Right: A quantum field theory.
  • Figure 3: Left: a path of $x(t)$ using ReLU as its activation. Right: a path using a step function.
  • Figure 4: A schematic picture of the relation between the NNFT and our formulation.
  • Figure 5: Left: the path-integral in the Euclidean time, denoted by the neural network with ReLU activation, (\ref{['xeuc']}). Right: the same path-integral with the activation function of the step function, (\ref{['xeuc2']}).
  • ...and 1 more figures