Analogy between Boltzmann machines and Feynman path integrals

arXiv cs.AI / 5/7/2026

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Key Points

The paper explains a detailed equivalence between Boltzmann machines used in machine learning and concepts from quantum statistical mechanics via Feynman’s path-integral description.
It argues that the hidden layers in Boltzmann machines (and related neural-network formalisms) can be viewed as discrete versions of path elements from the Feynman path-integral framework.
By treating computation as selecting and weighting combinations of “paths,” the work reframes learning as accumulating path weights that reproduce the correct input-to-output mapping.
As a key outcome, the authors provide general quantum circuit models that can serve as a common formalism for both Boltzmann-machine descriptions and path-integral descriptions.
The paper also connects this view to inverse quantum scattering problems, offering a more robust way to define “interpretable” hidden layers.

Abstract

We provide a detailed exposition of the connections between Boltzmann machines commonly utilized in machine learning problems and the ideas already well known in quantum statistical mechanics through Feynman's description of the same. We find that this equivalence allows the interpretation that the hidden layers in Boltzmann machines and other neural network formalisms are in fact discrete versions of path elements that are present within the Feynman path-integral formalism. Since Feynman paths are the natural and elegant depiction of interference phenomena germane to quantum mechanics, it appears that in machine learning, the goal is to find an appropriate combination of ``paths'', along with accumulated path-weights, through a network that cumulatively capture the correct

x \rightarrow y

map for a given mathematical problem. As a direct consequence of this analysis, we are able to provide general quantum circuit models that are applicable to both Boltzmann machines and to Feynman path integral descriptions. Connections are also made to inverse quantum scattering problems which allow a robust way to define ``interpretable'' hidden layers.