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arxiv: 2603.13073 · v2 · submitted 2026-03-13 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Exponential Scaling Barriers for Variational Quantum Eigensolvers

Manuel Hagelueken , David A. Kreplin , Florian Wieland , Marco F. Huber , Marco Roth
Authors on Pith no claims yet

Pith reviewed 2026-05-15 11:54 UTC · model grok-4.3

classification 🪐 quant-ph
keywords variational quantum eigensolveradaptive ansatzexponential scalingRényi entropyquantum chemistrymolecular simulationcircuit depthground state
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0 comments X

The pith

Adaptive VQE requires an exponentially growing number of iterations as the number of molecular orbitals increases.

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

The paper tests whether variational quantum eigensolvers can avoid the exponential cost of simulating large quantum systems by running adaptive VQE on more than twenty molecules whose active spaces range from four to ten orbitals. It finds that the number of adaptive iterations is predicted to high accuracy by the Rényi entropy computed from classical simulations, and that both the iteration count and the resulting circuit depth grow exponentially with system size. A reader should care because the original motivation for VQE was precisely to escape this exponential barrier on near-term hardware. If the observed scaling persists, current VQE forms cannot deliver high-fidelity results for large molecules without resources that grow exponentially.

Core claim

We demonstrate that the Rényi entropy derived from classical simulations predicts the required number of adaptive iterations of VQE with high accuracy (R² ≈ 0.99). We validate this on a benchmarking set of more than 20 different molecules with active spaces ranging from four to ten orbitals. For these molecules, we find an exponential scaling of the number of adaptive iterations, and in turn, of the circuit depth with the system size. We therefore conclude that it is unlikely that VQE in its current form is able to simulate large molecular systems with high fidelity without exponential resource requirements.

What carries the argument

The Rényi entropy computed on the classical wave function, which serves as a predictor of the number of adaptive VQE iterations needed to reach a given fidelity.

If this is right

  • Circuit depth in adaptive VQE grows exponentially with the number of orbitals.
  • The total quantum resources required scale exponentially rather than polynomially for larger molecular targets.
  • High-fidelity ground-state calculations on large molecules remain out of reach for VQE without exponential overhead.
  • Classical pre-computation of Rényi entropy can forecast the quantum cost before any quantum runs occur.

Where Pith is reading between the lines

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

  • Similar entropy-based diagnostics could be applied to other variational quantum algorithms to forecast their resource scaling.
  • The correlation suggests that any ansatz whose expressivity must match the target's entanglement will inherit comparable scaling limits.
  • Hybrid classical-quantum workflows might use the entropy predictor to decide when to switch from VQE to a different method.

Load-bearing premise

The exponential growth in iteration count seen for four-to-ten orbital active spaces will continue unchanged once classical simulation of the full system becomes impossible.

What would settle it

Performing adaptive VQE on a molecule whose active space has twenty or more orbitals and measuring whether the iteration count remains polynomial rather than exponential in orbital number.

Figures

Figures reproduced from arXiv: 2603.13073 by David A. Kreplin, Florian Wieland, Manuel Hagelueken, Marco F. Huber, Marco Roth.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic overview of the study. For each molecule in the benchmark set (Table [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. CEO-ADAPT-VQE with TETRIS extension applied to all molecules in Table [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Relationship between the Rényi entropy [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of (a) the number of ADAPT iterations [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Extrapolated number of ADAPT iterations [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of the number of ADAPT iterations [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of VQE variants for H [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Extrapolation of the number of ADAPT-VQE iter [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: shows the results: even at 10% of the full pool size, chemical accuracy is achieved, but nADAPT ap￾proximately doubles compared to the full pool. The rela￾tionship is monotonic. Smaller pools consistently require more iterations. This indicates that while reduced pools remain formally complete, they constrain the optimiza￾tion trajectory, forcing less direct paths to the ground state. A more striking examp… view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Correlation between the Rényi entropy [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Convergence diagnostics for hydrogen chains H [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Coefficient of determination (R [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Correlation between the Rényi entropy [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Extrapolation of per-iteration ADAPT-VQE circuit resources to [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
read the original abstract

The Variational Quantum Eigensolver (VQE) is widely regarded as a promising algorithm for calculating ground states of quantum systems that are intractable for classical computers. This promise is typically motivated by the hope of mitigating the exponential growth of Hilbert space with system size. Here we scrutinize how the computational cost of adaptive VQE scales with the size of the target system. We demonstrate that the R\'enyi entropy derived from classical simulations predicts the required number of adaptive iterations of VQE with high accuracy ($R^2 \approx 0.99$). We validate this on a benchmarking set of more than 20 different molecules with active spaces ranging from four to ten orbitals. For these molecules, we find an exponential scaling of the number of adaptive iterations, and in turn, of the circuit depth with the system size. We therefore conclude that it is unlikely that VQE in its current form is able to simulate large molecular systems with high fidelity without exponential resource requirements.

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

3 major / 1 minor

Summary. The manuscript reports numerical benchmarks of adaptive Variational Quantum Eigensolver (VQE) on more than 20 molecules with active spaces from 4 to 10 orbitals. It finds that the number of adaptive iterations scales exponentially with system size and is highly correlated (R² ≈ 0.99) with the Rényi entropy obtained from classical simulations. Based on this, the authors conclude that VQE in its current form is unlikely to simulate large molecular systems with high fidelity without requiring exponential resources.

Significance. If the observed exponential scaling and its correlation with Rényi entropy persist to larger systems, this would represent a significant finding that challenges the practical utility of VQE for quantum chemistry problems beyond small active spaces. The strength lies in the extensive benchmarking set and the independent computation of the predictor, providing a potential diagnostic tool. However, the significance is tempered by the limited range of system sizes tested.

major comments (3)
  1. [Results section (around the exponential fit and correlation analysis)] The exponential scaling is fitted to data from active spaces of 4-10 orbitals, but the manuscript does not provide error bars on the fit parameters or on the reported R² ≈ 0.99, making it difficult to assess the statistical robustness of the extrapolation claim.
  2. [Abstract and conclusion] The strong conclusion that VQE requires exponential resources for large systems rests on the assumption that the Rényi entropy-iteration correlation continues beyond the classically simulable regime; no scaling argument or additional evidence is given to support this continuity.
  3. [Methods or benchmarking set description] The criteria for selecting the more than 20 molecules and their active spaces are not detailed, which is necessary to evaluate whether the exponential scaling is general or specific to the chosen set.
minor comments (1)
  1. [Abstract] The abstract mentions 'high accuracy (R² ≈ 0.99)' but does not specify if this is for the correlation or the fit; clarify the exact quantity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We appreciate the referee's detailed review and constructive suggestions. Below we provide point-by-point responses to the major comments, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: The exponential scaling is fitted to data from active spaces of 4-10 orbitals, but the manuscript does not provide error bars on the fit parameters or on the reported R² ≈ 0.99, making it difficult to assess the statistical robustness of the extrapolation claim.

    Authors: We agree with this observation. In the revised version, we will compute and report error bars on the exponential fit parameters using bootstrap resampling and also provide the uncertainty on the R² value to better assess the robustness of our claims. revision: yes

  2. Referee: The strong conclusion that VQE requires exponential resources for large systems rests on the assumption that the Rényi entropy-iteration correlation continues beyond the classically simulable regime; no scaling argument or additional evidence is given to support this continuity.

    Authors: We acknowledge that our conclusions involve an extrapolation. While we cannot provide direct evidence for systems beyond classical simulability, the strong correlation with Rényi entropy, which is computable classically, provides a basis for the prediction. We will revise the abstract and conclusion to explicitly note this extrapolation and its basis in the observed data. revision: partial

  3. Referee: The criteria for selecting the more than 20 molecules and their active spaces are not detailed, which is necessary to evaluate whether the exponential scaling is general or specific to the chosen set.

    Authors: We will expand the Methods section to detail the criteria used for selecting the molecules and their active spaces, including considerations for diversity in molecular types and active space sizes to support the generality of the results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical correlation and explicit extrapolation remain independent of inputs.

full rationale

The paper computes Rényi entropy via separate classical simulations on the same small active spaces (4-10 orbitals) and reports an observed high-accuracy correlation (R² ≈ 0.99) with the measured number of adaptive VQE iterations across >20 molecules. This correlation is presented as an empirical validation rather than a definitional identity or a parameter fitted directly to the iteration counts and then relabeled as a prediction. The exponential scaling is fitted to the observed iteration counts versus system size, and the final claim about large-molecule intractability is framed as an extrapolation resting on the untested persistence of the entropy-iteration link outside the classically simulable regime. No load-bearing self-citation, uniqueness theorem, or ansatz smuggling is invoked; the derivation chain consists of direct computation, correlation measurement, and stated extrapolation without reducing any step to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on numerical observations from classical simulations of small molecules rather than new theoretical axioms or postulated entities.

axioms (1)
  • standard math Standard assumptions of quantum mechanics, variational principles, and the definition of Rényi entropy
    The VQE algorithm and entropy calculation presuppose these background results.

pith-pipeline@v0.9.0 · 5473 in / 1168 out tokens · 54538 ms · 2026-05-15T11:54:45.648028+00:00 · methodology

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