Quantum-assisted finite-element design optimization

Arne-Christian Voigt; Carsten Othmer; David Von Dollen; Dyon van Vreumingen; Florian Neukart; Michael Hartmann; Thomas B\"ack

REVIEW 3 major objections 4 minor 1 cited by

Quantum-assisted finite-element design optimization

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read An iterative QUBO optimizer on a quantum annealer reshapes a sphere to minimize a ray approximation of sound pressure at a microphone.

desk verdict New application, flawed core: Eq. (7) doesn't encode the stated loss, so the paper's central claim is unsupported; still worth a referee's time for the QUBO-FEM idea. read the letter →

arxiv 1908.03947 v1 pith:AGDY3LCJ submitted 2019-08-11 quant-ph

Dyon van Vreumingen , Florian Neukart , David Von Dollen , Carsten Othmer , Michael Hartmann , Arne-Christian Voigt , Thomas Bäck This is my paper · ORCID

classification quant-ph

keywords quantumannealingQUBOfinite-elementmethoddesignoptimizationsoundpressureraytracingshape

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces a quantum-assisted finite-element method for design optimization. It shows that a spherical mesh can be reshaped, iteration by iteration, so that a ray-tracing approximation of the sound pressure from an acoustic monopole at a microphone position is driven to zero. Each iteration encodes random candidate vertex movements as binary variables in a quadratic unconstrained binary optimization (QUBO) matrix, lets a quantum annealer pick one mutation per vertex by minimizing the matrix, and classically decodes the result into a new shape that still resembles the original design. The authors are explicit that the goal is to demonstrate how such a finite-element problem can be represented and solved on a QPU, not to outperform classical finite-element design optimization.

What carries the argument

The load-bearing object is the QUBO matrix $Q$ of Eq. (7), an $NK \times NK$ matrix whose entries pair mutations on neighboring vertices. For a mesh edge, the entry $Q_{i_1 j_1}^{i_2 j_2}$ collects, over the two simplices adjacent to the edge and over all K mutations of the off-edge third vertex, the partial ray-casting loss from selecting mutations $j_1$ and $j_2$ on the edge's vertices. A one-hot penalty, built from Eq. (8)-(9), is added to force exactly one selected mutation per vertex, and the QPU minimizes the quadratic form $x^T Q x$ over the resulting bitstring.

What would settle it

Run the algorithm's own ray-casting routine on a few hundred valid one-hot configurations of the same mesh. For each, compare the true total loss $L(S,c)$ with the QUBO value $x^T Q x$ from Eq. (7); the central claim holds only if the configuration with the smallest true loss is also the one with the smallest QUBO value.

Watch

Extended reading notes

Core claim

The central claim is that a shape-specific physical quantity—here, the number of sound rays that reflect off a surface and intersect a microphone plane—can be minimized by letting the QPU choose, for every mesh vertex, one of K random mutations in each iteration. The resulting shape keeps the general geometry of the starting sphere while developing features such as a sharp front tip that redirects rays around the microphone. The authors demonstrate this in two experiments: with the monopole at (2.5,0,0), the optimized shape sends no outgoing rays through the microphone plane; with the monopole at (0,3,2), two of the three loss-producing ray groups are eliminated and the remaining partial loss is reduced.

Load-bearing premise

The load-bearing assumption is that the quadratic matrix entries built from edge pairs, which average over all possible mutations of the third vertex of each triangle, equal the true ray-casting loss of the configuration actually selected; if the loss depends on which mutation that third vertex receives, the annealer is solving a different problem.

Editorial extensions

If this is right

In the first demonstration, with the source at (2.5,0,0), the optimized shape develops a sharp front tip and redirects every outgoing ray around the microphone plane, so the ray-count loss falls to zero.
When the source moves to (0,3,2) with a smaller step size, the routine removes two of the three loss-producing ray groups and reduces the remaining partial loss, with the residual attributed to a local optimum.
The same QUBO-driven vertex-selection routine can be pointed at other shape-specific quantities, such as vibration of individual elements, by replacing the partial-loss computation.
Because unlit rear triangles receive no rays, their QUBO entries vanish and the annealer moves them randomly; the paper notes that pruning such vertices would free qubits for more detailed shapes.
The method preserves the original design only at the level of vertex choices within a shrinking mutation radius; the shape remains recognizable while small convexity-preserving deformations accumulate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct numerical test not reported in the paper—comparing true ray-cast loss $L(S,c)$ with the Eq. (7) QUBO value over many one-hot configurations—would show whether the averaged pairwise matrix and the stated objective coincide; the averaging over the third vertex makes this worth checking.
The same formulation transfers to any ray-traceable objective, such as line-of-sight coverage, radiative heat load, or radar cross-section, since only the partial-loss oracle changes, not the QUBO structure.
One could restrict the annealer's vertex set to those whose adjacent simplices have nonzero partial loss, effectively implementing the paper's pruning suggestion and increasing usable mesh resolution on current hardware.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this is the first paper I've seen that encodes a finite-element shape-optimization problem as a QUBO and runs it on a D-Wave. That is a genuinely new application, and the authors deserve credit for framing the problem clearly, explaining the ray-casting simplification, and explicitly disclaiming any quantum speedup. The writing is honest about the 50-element embedding limit and the crudeness of the ray approximation. If you want an example of how an industrial team thinks about mapping a physical design problem to a QPU, this is a useful data point.

The soft spot is real and load-bearing. In Section 4.1, Eq. (7), the off-diagonal QUBO entry for an edge is defined as a sum over all K mutations of the third vertex of each adjacent simplex. But a one-hot configuration fixes exactly one mutation per vertex. So the QUBO objective is not the loss of the selected shape; it's an edge-averaged quantity that can be dominated by bad third-vertex mutations. The paper's own partition-function rewrite (Eqs. 4–6) legitimately sums over all configurations, but that doesn't justify using the same sum inside a single QUBO evaluation. The minimizer of the QUBO can differ from the minimizer of the true ray-loss. That invalidates the central claim as stated. The paper provides no numerical loss values, no error bars, and no baseline comparison, so you can't tell from the figures whether the QPU actually found lower-loss shapes or just different shapes.

A few smaller issues: beta and mu are tuned by trial and error on the same scenario used for evaluation, which weakens the reported success. No code or data are shipped, so the results are not independently checkable. The reference list leans heavily on the authors' own previous work, but not egregiously so for a niche application paper.

Is it salvageable? The underlying idea — decompose a mesh-deformation problem into vertex-mutation QUBOs and let an annealer pick the best combination — can work if Eq. (7) is fixed to use the selected third mutation, which likely requires a higher-order QUBO or a different decomposition. The paper would also need quantitative loss curves and a classical baseline (even a greedy hill-climber) to be convincing.

My verdict: the central derivation is wrong as written, so I would not cite this as evidence that quantum-assisted FEM works. But the problem is new, the exposition is accessible, and the flaw is subtle enough that a good referee could help the authors fix it. I'd send it to peer review with a request for major revision, not desk-reject it. A serious referee will find the Eq. (7) problem in an afternoon, and the authors seem capable of responding.

Referee Report

3 major / 4 minor

Summary. The manuscript introduces a quantum-assisted finite-element method for optimizing the shape of a 3D object against a scalar performance quantity, here the ray-approximated sound pressure at a microphone plane. The authors model sound propagation as straight rays with specular reflection, represent the object as a triangular mesh, and in each iteration generate K random candidate displacements ('mutations') per vertex. They define a loss function L(S,c) as the total over simplices of the fraction of rays hitting the microphone, encode the selection of one mutation per vertex as a QUBO with a one-hot penalty, and let a D-Wave 2000Q find the minimum. The shape is updated with the chosen mutations and the process repeats. The paper reports two experiments with qualitative figures and concludes that the algorithm successfully reduces the ray loss while preserving the general shape.

Significance. The paper's strength is its concrete end-to-end pipeline and its honest discussion of practical limitations: the local-coupling ray approximation, the limited number of elements on the QPU, and the random behavior of unilluminated regions. It also explicitly identifies the free parameters such as alpha, lambda, beta, mu, K, mesh resolution, and ray counts. If the QUBO objective faithfully represented L(S,c), the work would be a useful demonstration of a practical design-optimization encoding on a quantum annealer. However, because the central derivation in Sec. 4.1 substitutes a sum over third-vertex mutations for the selected mutation, the quantity minimized by the QPU is not shown to be L(S,c); combined with the absence of any numerical loss values, the empirical claim is not established. The significance of the result is therefore contingent on a repair that the paper does not provide.

major comments (3)

[Sec. 4.1, Eq. (7)] The QUBO matrix entry Q_{i_1 j_1}^{i_2 j_2} is defined as alpha times the sum over k=1..K of \hat l(s,j_1,j_2,k) for each simplex adjacent to the edge. For a feasible one-hot bitstring, the quadratic form evaluates this entry only for the single selected pair (j_1,j_2), so for a simplex with selected mutations j_a, j_b, j_c the objective receives alpha times [sum_k \hat l(s,j_a,j_b,k) + sum_k \hat l(s,j_a,j_c,k) + sum_k \hat l(s,j_b,j_c,k)]. This is not L(S,c(x)), which contains only \hat l(s,j_a,j_b,j_c) per simplex as in Eq. (3). The partition-function identity in Eq. (6) is valid because it sums over all K^3 third-vertex mutations, but minimization of the QUBO does not sum over alternatives; one good third mutation can be outweighed by K-1 poor ones. Unless \hat l is independent of the third mutation, which is not assumed or shown, the QPU minimizes a different objective than the stated ray-loss, so the central claim that the QPU minimizes sound pressure at the microphone is unsupported.
[Sec. 5, Figs. 4 and 5] The paper nowhere reports numerical values of L(S,c) before and after optimization, nor partial-loss totals, nor a convergence curve over iterations. The caption of Fig. 5(d) explicitly says the reduced loss is 'not shown in the figure.' Without quantitative data, the qualitative statement that the algorithm 'is successful in achieving its goal' cannot be checked, and it cannot be distinguished from random deformation or from minimization of the wrong objective identified above. The authors should provide measured loss values for the initial and final shapes and for each iteration.
[Sec. 5, hyperparameter selection] The values of beta and mu are selected by trial-and-error on the same scenario used for evaluation (Sec. 5: 'The values for beta and mu were chosen by trial-and-error search'), and no validation set or sensitivity analysis is provided. Given that the objective mismatch in Eq. (7) already prevents a claim of minimizing the ray loss, this tuning procedure further weakens the empirical evidence; the reported success could be specific to the displayed run rather than a robust property of the algorithm.

minor comments (4)

[Sec. 4.2, step 2] The soft convexity constraint rho_i is described only in a footnote and no formula or algorithm is given; this makes the step-size control and hence the mutation distribution irreproducible.
[Sec. 5] The mesh resolution (number of vertices) is not stated for either experiment; reporting N and the number of simplices is necessary for reproduction and for interpreting the claim about 'up to 50 elements'.
[Throughout] There are minor language errors, e.g., 'casted' should be 'cast' and 'a as quadratic' should be 'as a quadratic'; these do not affect the technical content.
[Sec. 4.1, Eq. (8)] Equation (8) should explicitly state that the expansion uses x_{ij} in {0,1}; otherwise the equality as written is not generally valid over the reals.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ray-casting loss is an externally defined objective, and the QUBO is explicitly constructed from it; no prediction is fitted and no load-bearing self-citation is used.

full rationale

The paper's central claim is that a QUBO formulation built from ray-casting partial losses can be minimized on a D-Wave QPU to reshape a mesh. The objective L(S,c) and the partial losses l(s,c) are defined by an external simulation (ray casting toward a microphone plane), not by the QUBO itself. The QUBO matrix in Eq. (7) is explicitly constructed from these partial losses, and the one-hot constraint of Eq. (8) is a standard penalty encoding, cited to the authors' prior traffic-flow work [15] only as a conventional construction technique, not as a load-bearing scientific premise. No uniqueness theorem from the same authors is invoked, and no parameter fitted to a subset of data is later renamed a prediction. The trial-and-error choice of beta and mu is hyperparameter tuning on the same demonstration scenario, but this is not circularity under the stated criteria because it does not reduce the claimed result to its own definition. The serious flaw in the paper is the mathematical mismatch in Eq. (7) identified in the reader's take: summing over all third-vertex mutations replaces the selected configuration's partial loss with an average, so the QUBO may minimize a different quantity than L(S,c). That is a correctness risk, not a circularity, because the objective is still externally defined. Accordingly, the circularity score is 0.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities; its variables are algorithmic choices (mutations, penalties, scaling factors). The key unpaid input is the incorrect pairwise decomposition of the loss into a QUBO matrix, which is listed as an axiom because the derivation does not prove it.

free parameters (7)

beta (mutation step-size control) = 0.7 for the first experiment, 0.3 for the second; tuned by trial and error over [0.3, 1.0]
Controls the maximum distance a vertex can move; the paper reports that too low or too high values degrade results, so the central demonstration depends on the chosen value.
mu (mutation radius decay exponent) = 0.18 in the first experiment; search range [0.15, 0.20]
Controls how quickly the mutation radius shrinks over iterations; tuned on the same experiments, and no independent validation is provided.
K (mutations per vertex) = 3
Sets the number of binary variables per vertex and the expressiveness of the search; all reported experiments use K=3.
lambda (one-hot penalty coefficient) = unspecified
Required to enforce exactly one mutation per vertex; no value is reported and feasibility of the QPU solution depends on it.
alpha (QUBO energy scaling) = unspecified
Introduced to absorb K^(N-3)/3; no value or sensitivity analysis is given.
number of rays per triangle = 50
Sampling resolution for partial losses; the paper notes variance can be reduced by more rays but does not test convergence.
mesh resolution (number of vertices) = not reported
The mesh is generated from a rectangular lattice in spherical coordinates, but the actual N is not specified; the embedding limit of about 50 elements is mentioned qualitatively.

assumptions (5)

domain assumption The D-Wave QPU returns a low-energy (ideally ground-state) solution to the submitted Ising/QUBO problem.
Invoked in Section 1, Eqs. (1) and (2); the entire method assumes the annealer solves the QUBO minimization reliably enough for iterative optimization.
ad hoc to paper Sound waves can be approximated as straight rays that reflect specularly off surfaces, with no wave effects or interference.
Stated as a major simplification in Section 4; if real acoustic behavior matters, the optimized shape may not reduce actual sound pressure.
ad hoc to paper The loss function L(S,c) can be minimized through the pairwise QUBO matrix in Eq. (7), which sums partial losses over all mutations of the third vertex.
Used in Section 4.1, Eq. (7); this assumption is mathematically incorrect for a one-hot assignment because the sum over k replaces the selected third mutation.
standard math A sufficiently large penalty lambda enforces exactly one mutation per vertex.
Used in Section 4.1, Eq. (9); this is a standard QUBO penalty construction, though the required value is not specified.
ad hoc to paper The soft convexity constraint rho_i, with beta <= 1, prevents chaotic shapes and approximately preserves convexity.
Defined in the footnote in Section 4.2; no formal proof or experiment validates that the constraint always holds.

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Cite this review

Pith. "Pith review of Quantum-assisted finite-element design optimization." pith.science (2026). https://pith.science/paper/AGDY3LCJ

@misc{pith2026190803947,
  author       = {Pith},
  title        = {Pith review of: Quantum-assisted finite-element design optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGDY3LCJ}},
  note         = {Machine review of arXiv:1908.03947}
}

read the original abstract

Quantum annealing devices such as the ones produced by D-Wave systems are typically used for solving optimization and sampling tasks, and in both academia and industry the characterization of their usefulness is subject to active research. Any problem that can naturally be described as a weighted, undirected graph may be a particularly interesting candidate, since such a problem may be formulated a as quadratic unconstrained binary optimization (QUBO) instance, which is solvable on D-Wave's Chimera graph architecture. In this paper, we introduce a quantum-assisted finite-element method for design optimization. We show that we can minimize a shape-specific quantity, in our case a ray approximation of sound pressure at a specific position around an object, by manipulating the shape of this object. Our algorithm belongs to the class of quantum-assisted algorithms, as the optimization task runs iteratively on a D-Wave 2000Q quantum processing unit (QPU), whereby the evaluation and interpretation of the results happens classically. Our first and foremost aim is to explain how to represent and solve parts of these problems with the help of a QPU, and not to prove supremacy over existing classical finite-element algorithms for design optimization.

Figures

Figures reproduced from arXiv: 1908.03947 by the authors.

**Figure 2.** A rigid sphere, which serves as the initial shape, and a rectangular area [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Initial setup for the first experiment with the monopole at (2 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Result after running the algorithm with the monopole at (2 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Results and comparison after executing the algorithm with the monopole [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 5.** Figure 5: figure 5. Here, we moved the monopole to (0 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

An Ising Machine Formulation for Design Updates in Topology Optimization of Flow Channels
cs.CE 2024-11 conditional novelty 6.0 of 10

Design updates in flow-channel topology optimization can be solved as a QUBO problem on an Ising machine, yielding fewer optimization steps but worse final energy dissipation than a classical method.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Estimation of eﬀective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning

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Smelyanskiy, Davide Venturelli, Alejandro Perdomo-Ortiz, Sergey Knysh, and Mark I

Vadim N. Smelyanskiy, Davide Venturelli, Alejandro Perdomo-Ortiz, Sergey Knysh, and Mark I. Dykman. Quantum Annealing via Environment-Mediated Quantum Diﬀusion. Physical Review Letters, 118(6), 2017

work page 2017

[3] [3]

Davide Venturelli, Dominic J. J. Marchand, and Galo Rojo. Quantum Annealing Implementation of Job-Shop Scheduling. jun 2015

work page 2015

[4] [4]

Zhang Jiang and Eleanor G. Rieﬀel. Non-commuting two-local Hamiltonians for quantum error suppression. Quantum Information Processing, 16(4), 2017

work page 2017

[5] [5]

Isakov, Guglielmo Mazzola, Vadim N

Sergei V. Isakov, Guglielmo Mazzola, Vadim N. Smelyanskiy, Zhang Jiang, Sergio Boixo, Hartmut Neven, and Matthias Troyer. Understanding quantum 14 tunneling through quantum Monte Carlo simulations. Physical Review Letters, 117(18), 2016

work page 2016

[6] [6]

O’Gorman, R

B. O’Gorman, R. Babbush, A. Perdomo-Ortiz, A. Aspuru-Guzik, and V. Smelyanskiy. Bayesian network structure learning using quantum annealing. The European Physical Journal Special Topics , 224(1):163–188, 2015

work page 2015

[7] [7]

Rieﬀel, Davide Venturelli, Bryan O’Gorman, Minh B

Eleanor G. Rieﬀel, Davide Venturelli, Bryan O’Gorman, Minh B. Do, Elicia M. Prystay, and Vadim N. Smelyanskiy. A case study in programming a quan- tum annealer for hard operational planning problems. Quantum Information Processing, 14(1), 2014

work page 2014

[8] [8]

Quantum optimization of fully connected spin glasses

Davide Venturelli, Salvatore Mandr` a, Sergey Knysh, Bryan O’Gorman, Rupak Biswas, and Vadim Smelyanskiy. Quantum optimization of fully connected spin glasses. Physical Review X, 5(3), 2015

work page 2015