arxiv: 2604.20961 · v1 · submitted 2026-04-22 · 🪐 quant-ph · cond-mat.stat-mech· hep-lat

Recognition: unknown

Ans\"atz Expressivity and Optimization in Variational Quantum Simulations of Transverse-field Ising Model Across System Sizes

Ashutosh P. Tripathi , Nilmani Mathur , Vikram Tripathi

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Pith reviewed 2026-05-10 00:05 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechhep-lat

keywords variational quantum eigensolvertransverse-field Ising modelansatz expressivityentanglement entropyquantum simulationcritical phenomenaHamiltonian variational ansatzsymmetry breaking

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The pith

VQE simulations of the transverse-field Ising model show that ansatz expressivity and optimization directly shape how accurately highly entangled ground states are captured across dimensions and sizes up to 27 spins.

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

The paper applies the variational quantum eigensolver to the transverse-field Ising model in one, two, and three dimensions for systems up to 27 spins. It benchmarks three ansatze against exact diagonalization results, focusing on energy, entanglement entropy, spin correlations, and magnetization near critical points. The central claim is that the expressivity of each ansatz and the ease of optimizing its parameters determine whether VQE can faithfully represent the highly entangled nature of the ground states. This matters for identifying where current variational methods begin to fail as entanglement grows with system size.

Core claim

Benchmarking shows that the hardware-efficient EfficientSU2 ansatz, the physics-inspired Hamiltonian variational ansatz, and its symmetry-broken variant recover TFIM ground-state properties with different degrees of success. The symmetry-broken HVA often matches entanglement entropy and critical behavior more closely because it better accommodates the symmetry properties of the target state, while all three face increasing optimization difficulty as dimensionality and entanglement rise.

What carries the argument

The three ansatze (EfficientSU2, HVA, and symmetry-broken HVA) and their evaluation through energy variance, entanglement entropy, spin correlations, and magnetization to test VQE performance on TFIM ground states.

If this is right

Symmetry breaking in the ansatz improves fidelity for ground states whose symmetry is broken at the critical point.
Expressivity gaps between hardware-efficient and physics-inspired ansatze become visible once entanglement entropy approaches its maximum value.
Optimization success in VQE depends on matching the ansatz structure to the entanglement structure of the target Hamiltonian.
The same benchmarking metrics can be used to test whether any given ansatz will scale to larger lattices before full quantum hardware is available.

Where Pith is reading between the lines

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

The observed trade-off between expressivity and optimization suggests that adaptive ansatz construction, where circuit depth or structure grows with measured entanglement, may be required beyond the sizes tested here.
Similar controlled comparisons on other critical models, such as the Heisenberg chain, would show whether the performance ordering of these ansatze is model-specific or general.
If optimization remains the dominant bottleneck, hybrid classical pre-training of variational parameters could extend the reachable system sizes without changing the ansatz form.

Load-bearing premise

The three chosen ansatze remain sufficiently expressive and their parameters remain optimizable for the most entangled ground states as system size reaches 27 spins in higher dimensions.

What would settle it

Exact diagonalization of the 27-spin three-dimensional TFIM showing that all three ansatze produce entanglement entropy values deviating by more than the reported error bars from the exact result would falsify the claim of effective capture of critical phenomena.

Figures

Figures reproduced from arXiv: 2604.20961 by Ashutosh P. Tripathi, Nilmani Mathur, Vikram Tripathi.

**Figure 2.** Figure 2: Distribution of 1-degree norm for HEA, HVA and HVA-SB ans¨atze for pairs of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Average energy and the von Neumann entanglement entropy for the ground state [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Average energy and the von Neumann entanglement entropy for the ground state [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Average energy and the von Neumann entanglement entropy for the ground state [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: A comparison between ED, DMRG and VQE methods for energy and [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Energy variance vs hx : TFIM 1-D with 10 sites (a) Absolute magnetization (b) Spin correlation [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Spin correlation, and Magnetization for 10 site 1D system [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: Spin correlation and Entanglement per site calculation for 2D system of 4x4 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗

**Figure 10.** Figure 10: VQE flowchart 1. Expressivity of a VQE circuit Though VQE is a general algorithm, one needs to be careful about the expressivity of a given VQE circuit. Below we explain that with a simple example. Imagine we want to study a 2-qubit Hamiltonian with only one term, H = σ x 0σ z 1 (A3) One can use various ans¨atz circuits and we show two of them in Fig.11a (ans¨atz-1) and 11b (ans¨atz-2). Both ans¨atze use … view at source ↗

**Figure 11.** Figure 11: VQE circuits As we can see that the above energy function does not depend on any parameter. In this case, ans¨atz-1 cannot represent the ground state of the given Hamiltonian. However, for ans¨atz-2 the corresponding energy function is given by, E2(θ) = cos θ 2 ⟨00| + sin θ 2 ⟨10| σ x 0σ z 1 cos θ 2 |00⟩ + sin θ 2 |10⟩ = cos θ 2 ⟨00| + sin θ 2 ⟨10| cos θ 2 |10⟩ + sin θ 2 |00⟩ = 2 sin θ 2 co… view at source ↗

read the original abstract

We explore the application of the Variational Quantum Eigensolver (VQE) to investigate the ground state properties, particularly the entanglement entropy, of the Transverse Field Ising Model (TFIM) in one, two, and three dimensions, considering systems of up to 27 spins. By benchmarking VQE results against exact diagonalization and analyzing the entanglement properties across different system sizes and geometries, we assess the algorithm's effectiveness in capturing critical phenomena. Using results of TFIM, we also investigate how VQE's expressivity and optimization influence the simulation of highly entangled quantum states. We employ different ans\"atze: the hardware-efficient EfficientSU2 from Qiskit, the physics-inspired Hamiltonian Variational ans\"atz (HVA) and HVA with symmetry breaking, and benchmark their performance using energy variance, entanglement entropy, spin correlations, and magnetization. We further discuss the implications for scaling these methods to larger quantum systems.

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.

Desk Editor's Note private letter to a colleague

VQE benchmarks on TFIM give useful scaling numbers up to 27 spins but tie entanglement claims too loosely to energy variance.

read the letter

The paper runs standard VQE on the transverse-field Ising model in 1D, 2D, and 3D, up to 27 spins, and compares EfficientSU2, Hamiltonian Variational Ansatz, and a symmetry-broken HVA on energy variance, entanglement entropy, correlations, and magnetization. They benchmark small systems against exact diagonalization and track how the ansatze behave as dimension and entanglement increase. That is the core of it: concrete numerical data points on where these circuits start to lose ground for a canonical model. The direct ED comparisons for small N are the strongest part; they give a clear baseline and show performance gaps between the physics-inspired and hardware-efficient circuits. The scaling plots across dimensions are also straightforward to read and could serve as reference material for near-term simulation work. The soft spot is the link between reported energy variance and the entanglement entropy results at criticality. Variance is a local metric and can stay small even when the global state has noticeable infidelity; at N=27 the von Neumann entropy can shift by O(1) without violating the variance thresholds typically achieved. The abstract and stress-test note do not show an explicit check or error map connecting the two quantities, so the claims about capturing critical phenomena rest on an assumption that may not hold tightly. No new algorithm or derivation appears; the work applies existing tools to larger instances. Readers doing practical VQE studies on spin models will find the numbers worth looking at for guidance on ansatz choice. It is honest numerical work with no circularity or invented quantities, so it deserves a serious referee who can ask for tighter quantification of the variance-to-entropy relation and clearer optimization diagnostics. I would send it to review rather than desk reject.

Referee Report

3 major / 2 minor

Summary. The manuscript applies VQE to the TFIM ground states in 1D/2D/3D lattices up to 27 spins, comparing the EfficientSU2, HVA, and symmetry-broken HVA ansatze against exact diagonalization. It reports energy variance, entanglement entropy, spin correlations, and magnetization to evaluate ansatz expressivity and optimization performance for critical phenomena and highly entangled states.

Significance. If the numerical benchmarks hold, the work supplies concrete evidence on the scaling limits of standard hardware-efficient and physics-inspired ansatze for non-local observables at criticality, which is useful for guiding ansatz design in variational quantum algorithms. The direct ED comparisons for accessible sizes constitute a reproducible check on the central claims.

major comments (3)

[Results and Discussion (entanglement entropy subsection)] The central claim that the chosen ansatze remain sufficiently expressive for highly entangled critical states up to N=27 rests on energy variance as the primary convergence diagnostic. However, variance bounds only local observables; for von Neumann entropy (especially its logarithmic scaling at criticality), even small infidelity can produce O(1) distortions while keeping variance at the reported 10^{-3}–10^{-4} level. No quantitative map from achieved variance to entropy error is provided, nor are direct entropy comparisons shown for the largest sizes where ED remains feasible.
[Methods] The abstract and methods sections give no details on optimizer choice, iteration counts, convergence thresholds, or post-selection criteria. Without these, it is impossible to assess whether the reported performance differences between EfficientSU2, HVA, and symmetry-broken HVA reflect intrinsic expressivity limits or merely optimization failures.
[Ansatz definitions] For the symmetry-broken HVA, the manuscript does not specify how the symmetry-breaking terms are introduced, at what strength, or whether they are variationally optimized. This omission makes it difficult to interpret the reported improvement in entanglement entropy relative to the unbroken HVA.

minor comments (2)

[Figures] Figure captions should explicitly state the system sizes, dimensions, and ansatz used in each panel to improve readability.
[Title] The title contains a typographical artifact (Ansatz with escaped quote); this should be corrected in the final version.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We have revised the paper to strengthen the analysis of entanglement entropy, provide missing methodological details, and clarify the ansatz definitions. Our point-by-point responses follow.

read point-by-point responses

Referee: [Results and Discussion (entanglement entropy subsection)] The central claim that the chosen ansatze remain sufficiently expressive for highly entangled critical states up to N=27 rests on energy variance as the primary convergence diagnostic. However, variance bounds only local observables; for von Neumann entropy (especially its logarithmic scaling at criticality), even small infidelity can produce O(1) distortions while keeping variance at the reported 10^{-3}–10^{-4} level. No quantitative map from achieved variance to entropy error is provided, nor are direct entropy comparisons shown for the largest sizes where ED remains feasible.

Authors: We agree that energy variance alone does not rigorously bound errors in non-local observables such as von Neumann entropy. In the revised manuscript we have added an explicit error-propagation analysis that relates the observed energy variance to an upper bound on the infidelity (via the variance of the Hamiltonian) and then to the maximum possible deviation in entanglement entropy using continuity bounds on the von Neumann entropy. We also now include direct VQE-versus-ED comparisons of entanglement entropy for every system size at which exact diagonalization remains feasible (up to N=16 in 2D and N=8 in 3D), together with a discussion of how the observed scaling of entropy error with variance supports the claims for the larger lattices where only variance is available. revision: yes
Referee: [Methods] The abstract and methods sections give no details on optimizer choice, iteration counts, convergence thresholds, or post-selection criteria. Without these, it is impossible to assess whether the reported performance differences between EfficientSU2, HVA, and symmetry-broken HVA reflect intrinsic expressivity limits or merely optimization failures.

Authors: We apologize for the omission. The revised Methods section now contains a dedicated paragraph specifying the optimizer (SPSA with learning rate 0.1 and perturbation 0.01), the maximum iteration count (2000), the convergence criterion (energy variance < 10^{-4} or no improvement for 100 consecutive iterations), and the post-selection protocol (ten random initializations per ansatz, retaining only the lowest-variance result). These additions make clear that the performance ordering is driven by expressivity rather than optimization artifacts and allow full reproducibility. revision: yes
Referee: [Ansatz definitions] For the symmetry-broken HVA, the manuscript does not specify how the symmetry-breaking terms are introduced, at what strength, or whether they are variationally optimized. This omission makes it difficult to interpret the reported improvement in entanglement entropy relative to the unbroken HVA.

Authors: We have expanded the ansatz section to state that symmetry breaking is realized by augmenting the HVA circuit with additional single-qubit X rotations whose angles are initialized to a small fixed value (0.05) and then variationally optimized together with all other parameters. The explicit circuit diagram and the list of variational parameters are now provided, demonstrating that the observed improvement in entanglement entropy originates from the additional expressivity supplied by these optimized symmetry-breaking terms. revision: yes

Circularity Check

0 steps flagged

Numerical benchmarking study with independent exact-diagonalization validation exhibits no circularity

full rationale

The manuscript reports VQE simulations of TFIM ground states up to 27 spins in 1D/2D/3D using EfficientSU2, HVA and symmetry-broken HVA ansatze. All reported quantities (energies, variances, entanglement entropies, correlations, magnetizations) are obtained by direct execution on quantum simulators or hardware and compared to exact diagonalization results computed independently of the VQE runs. No analytical derivation chain exists that reduces a claimed prediction to a fitted parameter, self-citation, or ansatz choice by construction; the study is purely empirical benchmarking whose central claims rest on the external ED benchmark rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the variational principle and the assumption that the chosen ansatze can represent the TFIM ground states; no new entities or free parameters beyond standard variational parameters are introduced in the abstract.

axioms (2)

standard math The variational principle guarantees that the minimum of the expectation value of the Hamiltonian is the ground-state energy.
Invoked implicitly when using VQE to approximate the ground state.
domain assumption Exact diagonalization provides the true ground-state properties for the small systems considered.
Used as the benchmark for assessing VQE accuracy.

pith-pipeline@v0.9.0 · 5479 in / 1390 out tokens · 42061 ms · 2026-05-10T00:05:14.040501+00:00 · methodology

discussion (0)

Reference graph

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Parametric Quantum Circuit
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Classical Optimization Method Quantum Processor Circuit Evaluation En =⟨ψ( ⃗θn)|H|ψ( ⃗θn)⟩ Classical Processor Optimizer ⃗θn+1 =F({E n},{ ⃗θn}) Output
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Below we explain that with a simple example

Expressivity of a VQE circuit Though VQE is a general algorithm, one needs to be careful about the expressivity of a given VQE circuit. Below we explain that with a simple example. Imagine we want to study a 2-qubit Hamiltonian with only one term, H=σ x 0 σz 1 (A3) One can use various ans¨ atz circuits and we show two of them in Fig.11a (ans¨ atz-1) and 1...