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arxiv: 2301.06217 · v1 · submitted 2023-01-15 · 🪐 quant-ph · cs.AI· cs.LG

Analogy between Boltzmann machines and Feynman path integrals

Srinivasan S. Iyengar , Sabre Kais This is my paper

Pith reviewed 2026-05-24 10:06 UTC · model grok-4.3

classification 🪐 quant-ph cs.AIcs.LG
keywords Boltzmann machinesFeynman path integralsquantum circuitsmachine learninghidden layersquantum statistical mechanicsinverse quantum scattering
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The pith

Hidden layers in Boltzmann machines correspond to discrete versions of Feynman path elements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an equivalence between Boltzmann machines used in machine learning and the Feynman path-integral formalism from quantum statistical mechanics. This equivalence supports the view that hidden layers function as discrete path elements, so that machine learning amounts to selecting combinations of such paths together with their accumulated weights to produce the desired input-to-output map. A sympathetic reader would care because the same structure then supplies quantum circuit models that apply uniformly to both Boltzmann machines and path integrals. The analysis further connects the hidden layers to inverse quantum scattering problems as a route to interpretability.

Core claim

The formal connections between Boltzmann machines and Feynman's description of quantum statistical mechanics allow the interpretation that the hidden layers in Boltzmann machines and other neural network formalisms are discrete versions of path elements present within the Feynman path-integral formalism. Feynman paths thereby supply a natural depiction of interference phenomena, implying that machine learning seeks an appropriate combination of paths along with accumulated path-weights through a network that cumulatively capture the correct mapping. As a direct consequence, general quantum circuit models applicable to both Boltzmann machines and Feynman path integral descriptions can be set,

What carries the argument

The path-element interpretation that equates hidden layers in Boltzmann machines with discrete Feynman path elements and thereby yields quantum circuit models.

Load-bearing premise

The formal structures of Boltzmann machines and Feynman path integrals are sufficiently isomorphic that the path-element interpretation and derived quantum-circuit models follow directly.

What would settle it

A concrete calculation on a small Boltzmann machine showing that its probability distribution over configurations does not match the distribution obtained by summing the corresponding discrete Feynman paths with the same weights.

Figures

Figures reproduced from arXiv: 2301.06217 by Sabre Kais, Srinivasan S. Iyengar.

Figure 1
Figure 1. Figure 1: FIG. 1. RBM network G=(V,E) showing the biases [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Explicit version of Figure [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Neural network depiction of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Illustration of the approach in Section [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. 3-local Neural network depiction with entropy in Eq. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript provides an expository analogy between Boltzmann machines (and related neural networks) and Feynman path integrals in quantum statistical mechanics. It interprets hidden layers as discrete versions of path elements, argues that machine learning seeks appropriate combinations of such paths with accumulated weights to realize x→y maps, and derives general quantum circuit models applicable to both formalisms, with additional links to inverse quantum scattering for interpretability of hidden units.

Significance. If the structural mapping can be made rigorous, the work could usefully bridge classical machine-learning architectures with quantum path-integral techniques, potentially motivating quantum-circuit realizations of probabilistic models and new interpretive tools for neural-network layers. The paper is credited for explicitly connecting the two domains and for highlighting the inverse-scattering route to interpretability; however, because the central step remains interpretive rather than a derivation with controlled approximations or numerical checks, the immediate technical impact is modest.

major comments (2)
  1. [Abstract] Abstract and the paragraph introducing the path-element interpretation: the claim that hidden layers 'are in fact discrete versions of path elements' is asserted to follow directly from the equivalence, yet the manuscript does not supply an explicit isomorphism that maps the real Boltzmann factor exp(−E) onto the complex Feynman amplitude exp(iS/ℏ) while preserving interference phases; without this step the subsequent quantum-circuit constructions risk capturing only classical summation.
  2. [Quantum circuit models] Section presenting the general quantum circuit models: these models are presented as immediate consequences of the analogy, but the text does not demonstrate how real-valued RBM weights are promoted to complex path weights or how the discrete-path discretization avoids uncontrolled errors in the dynamics; the skeptic concern that the mapping may omit phases therefore remains unaddressed and is load-bearing for the claimed equivalence.
minor comments (2)
  1. Notation for the energy/action functionals is introduced without a consolidated table comparing the Boltzmann and Feynman conventions side-by-side; this would improve readability.
  2. [Abstract] The abstract states that the analysis 'allows' the path-element interpretation but does not distinguish this interpretive step from any new theorem or derivation; a single clarifying sentence would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and for noting the expository nature of the work. The manuscript draws a structural analogy between the two formalisms rather than claiming a rigorous isomorphism. We address each major comment below and will revise the manuscript to clarify the interpretive character of the connection.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the paragraph introducing the path-element interpretation: the claim that hidden layers 'are in fact discrete versions of path elements' is asserted to follow directly from the equivalence, yet the manuscript does not supply an explicit isomorphism that maps the real Boltzmann factor exp(−E) onto the complex Feynman amplitude exp(iS/ℏ) while preserving interference phases; without this step the subsequent quantum-circuit constructions risk capturing only classical summation.

    Authors: We agree that no explicit isomorphism is supplied that maps real Boltzmann factors onto complex amplitudes while preserving phases. The manuscript presents the link as an expository analogy that permits interpreting hidden units as discrete path elements; it does not assert a direct mathematical equivalence that incorporates quantum interference. The quantum-circuit constructions follow from this structural view but remain illustrative. We will revise the abstract and the relevant introductory paragraph to state explicitly that the connection is interpretive rather than a phase-preserving isomorphism, thereby removing any implication of a stronger equivalence. revision: yes

  2. Referee: [Quantum circuit models] Section presenting the general quantum circuit models: these models are presented as immediate consequences of the analogy, but the text does not demonstrate how real-valued RBM weights are promoted to complex path weights or how the discrete-path discretization avoids uncontrolled errors in the dynamics; the skeptic concern that the mapping may omit phases therefore remains unaddressed and is load-bearing for the claimed equivalence.

    Authors: The referee correctly observes that the manuscript does not demonstrate promotion of real RBM weights to complex path weights, nor does it analyze discretization errors or phase factors. Because the paper focuses on the conceptual analogy rather than a quantitative derivation, these technical steps are absent. We will add a clarifying paragraph to the quantum-circuit section that acknowledges these limitations, states that the models are motivated by the analogy but do not constitute a controlled mapping, and notes that phases are not automatically retained. revision: yes

Circularity Check

0 steps flagged

No circularity: interpretive analogy between independent formalisms

full rationale

The paper advances an interpretive mapping: hidden layers in Boltzmann machines are viewed as discrete versions of Feynman path elements. The abstract and reader's summary indicate this follows from structural equivalence of the two formalisms without introducing fitted parameters, self-referential equations, or load-bearing self-citations that reduce the central claim to its own inputs. No derivation step equates a prediction to a fitted quantity by construction, and the real-vs-complex distinction raised by the skeptic is a correctness concern rather than a circularity reduction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the structural similarity between the two partition-function formalisms; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption The partition function of a Boltzmann machine can be identified with a discretized Feynman path integral without loss of the essential probabilistic interpretation.
    Invoked in the abstract when the hidden-layer interpretation is asserted.

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Reference graph

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