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arxiv: 2502.04271 · v3 · pith:IITALK2Snew · submitted 2025-02-06 · 🪐 quant-ph · cs.LG

Variational decision diagrams for quantum-inspired machine learning applications

Vladimir Vargas-Calder\'on , Santiago Acevedo-Mancera , Herbert Vinck-Posada This is my paper

Pith reviewed 2026-05-23 03:45 UTC · model grok-4.3

classification 🪐 quant-ph cs.LG
keywords variational decision diagramsquantum machine learningbarren plateausground state estimationIsing HamiltonianHeisenberg Hamiltoniandecision diagrams
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The pith

Variational decision diagrams represent quantum states and show no barren plateaus when trained on ground states of Ising and Heisenberg Hamiltonians.

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

The paper introduces variational decision diagrams as a graph-based structure that merges the redundancy-handling of decision diagrams with variational parameters to represent quantum states efficiently. It tests these on the task of finding ground states for two common spin Hamiltonians and measures the variance of gradients during training. The analysis finds no evidence of vanishing gradients, indicating that the structures remain trainable. This suggests decision diagrams could serve as an alternative to circuit-based ansatze in quantum machine learning.

Core claim

Variational decision diagrams (VDDs) are a new graph structure for representing quantum states that combines decision diagrams with variational methods. When applied to ground state estimation for transverse-field Ising and Heisenberg Hamiltonians, gradient variance analysis shows no signs of vanishing gradients or barren plateaus, indicating that VDDs are trainable.

What carries the argument

Variational decision diagrams (VDDs), graph structures that exploit data redundancies in quantum states while incorporating variational parameters for adaptability.

If this is right

  • VDDs provide an efficient alternative to variational quantum circuits for representing states in QML.
  • Training VDDs is feasible for ground state problems without gradient vanishing issues in the tested cases.
  • The structural benefits of decision diagrams can be adapted for variational optimization in quantum-inspired learning.
  • Insights from VDDs may guide the design of new ansatze that avoid barren plateaus.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar gradient analysis could be applied to other quantum-inspired models to check trainability.
  • VDDs might extend to problems beyond ground state estimation, such as classification tasks.
  • If scalable, VDDs could reduce the resource requirements compared to full quantum circuit simulations.

Load-bearing premise

The absence of barren plateaus observed for the transverse-field Ising and Heisenberg Hamiltonians implies the same for VDDs in general.

What would settle it

Finding vanishing gradients or barren plateaus when applying VDDs to ground state estimation of a different Hamiltonian or at larger system sizes.

Figures

Figures reproduced from arXiv: 2502.04271 by Herbert Vinck-Posada, Santiago Acevedo-Mancera, Vladimir Vargas-Calder\'on.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic layout of the VDD accordion ansatz for [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Gradient variances, computed with eq. (9), averaged over random values of some parameters of the accordion ansatz [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Percentual energy error curves as a function of the epoch number for a system with 10 qubits for the following [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

Decision diagrams (DDs) have emerged as an efficient tool for simulating quantum circuits due to their capacity to exploit data redundancies in quantum states and quantum operations, enabling the efficient computation of probability amplitudes. However, their application in quantum machine learning (QML) has remained unexplored. This paper introduces variational decision diagrams (VDDs), a novel graph structure that combines the structural benefits of DDs with the adaptability of variational methods for efficiently representing quantum states. We investigate the trainability of VDDs by applying them to the ground state estimation problem for transverse-field Ising and Heisenberg Hamiltonians. Analysis of gradient variance suggests that training VDDs is possible, as no signs of vanishing gradients--also known as barren plateaus--are observed. This work provides new insights into the use of decision diagrams in QML as an alternative to design and train variational ans\"atze.

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 paper introduces variational decision diagrams (VDDs), a graph-based structure that merges decision diagrams' redundancy exploitation with variational parameters for representing quantum states. It applies VDDs to the ground-state estimation task for the transverse-field Ising and Heisenberg Hamiltonians and reports that gradient-variance analysis shows no vanishing gradients (barren plateaus), concluding that VDDs are trainable and offering a new alternative to conventional variational ansätze in quantum machine learning.

Significance. If the reported absence of barren plateaus generalizes, VDDs could provide an efficient, redundancy-aware alternative to standard parameterized quantum circuits for QML tasks, potentially improving both simulation efficiency and trainability on structured problems. The work is credited for exploring decision diagrams in a variational QML setting, an underexplored direction.

major comments (2)
  1. [Abstract] Abstract (and results section on gradient variance): the central claim that 'training VDDs is possible' because 'no signs of vanishing gradients are observed' rests exclusively on measurements for the transverse-field Ising and Heisenberg models. No general bound on gradient variance, scaling analysis with system size or parameter count, or tests on other Hamiltonians (e.g., 2D lattices, long-range interactions) are supplied, so the extrapolation from these two 1D cases to general trainability is unsupported.
  2. [Abstract] The weakest assumption identified in the stress-test note is load-bearing: the manuscript treats non-vanishing variance on two specific structured 1D spin chains as evidence that VDDs avoid barren plateaus in general, yet nothing in the supplied analysis rules out variance collapse for different connectivities or larger variational parameter spaces.
minor comments (2)
  1. Notation for the VDD graph structure and the precise definition of the variational parameters should be introduced with an explicit diagram or pseudocode early in the manuscript to aid reproducibility.
  2. The abstract states that DDs enable 'efficient computation of probability amplitudes' but does not quantify the complexity gain relative to standard tensor-network or circuit simulators for the VDD case; a brief complexity statement would strengthen the motivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on the scope of our claims. We address each major point below and agree that revisions are needed to clarify the limited scope of the reported results.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and results section on gradient variance): the central claim that 'training VDDs is possible' because 'no signs of vanishing gradients are observed' rests exclusively on measurements for the transverse-field Ising and Heisenberg models. No general bound on gradient variance, scaling analysis with system size or parameter count, or tests on other Hamiltonians (e.g., 2D lattices, long-range interactions) are supplied, so the extrapolation from these two 1D cases to general trainability is unsupported.

    Authors: We agree that the gradient-variance results are confined to the transverse-field Ising and Heisenberg models and that no general bound, scaling analysis, or tests on other Hamiltonians are provided. The manuscript therefore does not support a claim of general trainability. We will revise the abstract and the relevant results section to state explicitly that no signs of vanishing gradients were observed for these two specific 1D models, removing any implication of broader applicability. revision: yes

  2. Referee: [Abstract] The weakest assumption identified in the stress-test note is load-bearing: the manuscript treats non-vanishing variance on two specific structured 1D spin chains as evidence that VDDs avoid barren plateaus in general, yet nothing in the supplied analysis rules out variance collapse for different connectivities or larger variational parameter spaces.

    Authors: The analysis is restricted to the two cited 1D spin chains; nothing in the work rules out variance collapse under different connectivities or larger parameter spaces. We will revise the abstract and discussion to make this limitation explicit and to avoid any suggestion that the results establish avoidance of barren plateaus in general. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical gradient-variance measurements on two Hamiltonians presented directly as evidence

full rationale

The paper's central claim rests on explicit computation of gradient variance for VDDs applied to ground-state estimation of the transverse-field Ising and Heisenberg models, with the observation that variance does not vanish. This is a direct empirical measurement rather than a derivation that reduces by construction to a fitted parameter, self-defined quantity, or self-citation chain. No equations, ansatzes, or uniqueness theorems are invoked that loop back to the paper's own inputs. The limitation to two specific 1D Hamiltonians affects the strength of the generalization but does not constitute circularity under the defined criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities beyond the VDD structure itself are stated.

invented entities (1)
  • Variational decision diagrams (VDDs) no independent evidence
    purpose: Graph structure that combines decision diagrams with variational parameters to represent quantum states for machine learning
    New object introduced in the paper; no independent evidence supplied in the abstract.

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    Updating variational parameters To close the circle, we are now able to update the pa- rameters in a convenient way so as to minimise the ex- pected value of the HamiltonianH. This can be achieved by several gradient descent algorithms [54], the simpler of which can be given by the stochastic gradient descent update rule: θj ←θ j −η ∂⟨H⟩ ∂θj ,(A7) where t...

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