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arxiv: 2403.11995 · v1 · submitted 2024-03-18 · 🪐 quant-ph

Hamiltonian-reconstruction distance as a success metric for the Variational Quantum Eigensolver

Leo Joon Il Moon , Mandar M. Sohoni , Michael A. Shimizu , Praveen Viswanathan , Kevin Zhang , Eun-Ah Kim , Peter L. McMahon This is my paper

Pith reviewed 2026-05-24 03:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords variational quantum eigensolverhamiltonian reconstructionground statefidelityising modelquantum simulationsuccess metricnoisy quantum computing
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The pith

Hamiltonian-reconstruction distance indicates whether VQE has reached the ground state without knowing the ground state or energy.

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

The paper shows that a distance metric derived from Hamiltonian reconstruction can assess how close a VQE solution is to the true ground state. This works by seeing how much the target Hamiltonian differs from one reconstructed assuming the variational state is an eigenstate. A reader would care because VQE runs on current hardware risk stopping too soon when the energy stops decreasing but the state is still not optimal. Tests on 1D transverse-field Ising with 11 qubits and 2D J1-J2 Ising with 6 qubits, both in simulation and on a trapped-ion computer, confirm the distance tracks fidelity to the ground state and correctly signals to continue when energy plateaus.

Core claim

Numerical simulations and demonstrations on a cloud-based trapped-ion quantum computer show that for one-dimensional transverse-field-Ising models with 11 qubits and two-dimensional J1-J2 transverse-field-Ising models with 6 qubits, the Hamiltonian-reconstruction distance indicates whether VQE has found the ground state or not and has a useful correlation with the fidelity between the VQE solution and the true ground state, including in cases where energy plateaus.

What carries the argument

Hamiltonian-reconstruction distance obtained by inferring a Hamiltonian from a candidate state and measuring its discrepancy with the target Hamiltonian

If this is right

  • Enables quality assessment of VQE solutions without access to the true ground state.
  • Prevents erroneous early termination of VQE when energy has plateaued.
  • Exhibits correlation with state fidelity in the studied models.
  • Applies to both classical simulations and experiments on noisy trapped-ion processors.

Where Pith is reading between the lines

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

  • This approach may generalize to other variational methods for finding eigenstates in quantum systems.
  • It could be integrated into VQE workflows to provide a more reliable stopping criterion on real hardware.
  • Further validation on larger qubit numbers or different Hamiltonian types would strengthen confidence in its broad utility.

Load-bearing premise

The Hamiltonian reconstruction remains reliable and the distance informative when applied to the noisy states produced by VQE on the tested Ising models, even after the energy has plateaued.

What would settle it

An experiment or simulation where the reconstruction distance is small for a VQE state that has low fidelity to the actual ground state would show the metric is not dependable.

Figures

Figures reproduced from arXiv: 2403.11995 by Eun-Ah Kim, Kevin Zhang, Leo Joon Il Moon, Mandar M. Sohoni, Michael A. Shimizu, Peter L. McMahon, Praveen Viswanathan.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: Similarly, Figure 2c shows the energy, HR distance, and fidelity in simulation as a function of the VQE [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: An example of a scenario, in simulation, of an 11-qubit 1D transverse-field-Ising model VQE optimization, where the HR distance suggests that the VQE solution isn’t optimal even though it appears to have converged. We demonstrate a scenario where measuring the HR distance does help in diagnosing a VQE solution. We simulated VQE for an 11-qubit 1D-TFIM Hamiltonian with J = 0.5 (S7) without shot noise and de… view at source ↗
Figure 2
Figure 2. Figure 2: HR distances and fidelities of 100 randomly perturbed ground state wavefunctions for different sets of operators for Hamiltonian reconstruction. The red star in each plot indicates the fidelity and the HR distance of the ground state. Each subplot uses the following set of operators for Hamiltonian reconstruction: a. {X, ZZ}, b. {X, ZZ, Z}, c. {X, ZZ, Y }, d. {X, ZZ, XX}, e. {X, ZZ, Y Y }, f. {X, ZZ, Z, Y … view at source ↗
Figure 3
Figure 3. Figure 3: 11-qubit 3 layer ALA ansatz Ry Ry Ry Ry Ry Ry H H H H H H Ry Ry Ry Ry Ry Ry Ry Ry Ry Ry Ry Ry [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Depiction of YY ansatz. a. 4-qubit 1 layer YY ansatz. b. 11-qubit 1 layer YY ansatz, generalized from the 4-qubit 1 layer YY ansatz [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: HR distance and Hamiltonian variance measured with IonQ device using lists of parameters sampled from the simulated VQE for the 11-qubit 1D-TFIM Hamiltonian. Both subplots further show how energy measured with IonQ device changes with simulated VQE iterations. We sampled the lists of parameters from the simulated VQE and measured the HR distances (Hamiltonian variances) and energies with an IonQ’s device f… view at source ↗
Figure 7
Figure 7. Figure 7: HR distance, Hamiltonian variance, and energy measured with IonQ device using the list of parameters sampled from the simulated VQE for the 6-qubit J1-J2 TFIM Hamiltonian. Color conventions are same as the Appendix [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: HR distance and fidelity measured for the 11-qubit 1D-TFIM (a) and for the 6-qubit J1-J2 TFIM (b), while incrementally increasing the depolarization noise. For both models, we see negative correlations between HR distances and fidelities as we vary depolarization noise. However, we further see how a certain value of fidelity does not necessarily correspond to a certain value of HR distance across different… view at source ↗
Figure 9
Figure 9. Figure 9: HR distance and Fidelity measured for randomly perturbed wavefunctions described in Appendix C using different values of depolarization noise for the 11 qubit 1D-TFIM. The depolarization noise value for each of the subplot is listed as p. When the probability of depolarization noise is more than 0.1 (p > 0.1), we start to lose the negative correlation between the HR distance and the fidelity drastically. W… view at source ↗
Figure 10
Figure 10. Figure 10: Dependence of the HR distance on the number of shots used in measurements for the 11-qubit 1D-TFIM and the 6-qubit J1-J2 TFIM. All simulations were run using the ALA ansatzes in Appendix E with depolarization noise values equal to the reported average IONQ harmony gate infidelities. Appendix [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: VQE ran with a) J = 0.5 and b) J = 1 for 8 qubit 1-D Transverse Field Ising model. Blue points are HR distances measured for each iteration of VQE optimization. Red points are fidelities between the output state from each iteration of VQE optimization and the ground state. Lastly, orange points are fidelities between the output state from each iteration of VQE optimization and the first excited state [PI… view at source ↗
Figure 12
Figure 12. Figure 12: Hamiltonian variance (⟨(∆H) 2 ⟩) from both experiment (with IonQ device) and simulation, using the list of parameters from the simulated VQE. A 14-point moving average window was used to better show the correlation between the Hamiltonian variance and the fidelity. a) simulated Hamiltonian (light-blue circle) variance and Hamiltonian variance measured with IonQ device (light-blue triangle) for the 11-qubi… view at source ↗
read the original abstract

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm for quantum simulation that can be run on near-term quantum hardware. A challenge in VQE -- as well as any other heuristic algorithm for finding ground states of Hamiltonians -- is to know how close the algorithm's output solution is to the true ground state, when the true ground state and ground-state energy are unknown. This is especially important in iterative algorithms, such as VQE, where one wants to avoid erroneous early termination. Recent developments in Hamiltonian reconstruction -- the inference of a Hamiltonian given an eigenstate -- give a metric can be used to assess the quality of a variational solution to a Hamiltonian-eigensolving problem. This metric can assess the proximity of the variational solution to the ground state without any knowledge of the true ground state or ground-state energy. In numerical simulations and in demonstrations on a cloud-based trapped-ion quantum computer, we show that for examples of both one-dimensional transverse-field-Ising (11 qubits) and two-dimensional J1-J2 transverse-field-Ising (6 qubits) spin problems, the Hamiltonian-reconstruction distance gives a helpful indication of whether VQE has yet found the ground state or not. Our experiments included cases where the energy plateaus as a function of the VQE iteration, which could have resulted in erroneous early stopping of the VQE algorithm, but where the Hamiltonian-reconstruction distance correctly suggests to continue iterating. We find that the Hamiltonian-reconstruction distance has a useful correlation with the fidelity between the VQE solution and the true ground state. Our work suggests that the Hamiltonian-reconstruction distance may be a useful tool for assessing success in VQE, including on noisy quantum processors in practice.

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

1 major / 0 minor

Summary. The manuscript claims that the Hamiltonian-reconstruction distance, obtained via a separate reconstruction procedure from a variational state, provides a practical metric for assessing proximity to the true ground state in VQE without knowledge of the ground-state energy or wavefunction. Numerical simulations and trapped-ion hardware experiments on 1D transverse-field Ising (11 qubits) and 2D J1-J2 transverse-field Ising (6 qubits) models demonstrate that this distance correlates with ground-state fidelity and correctly indicates the need to continue optimization in cases where the variational energy has plateaued, including for noisy variational states.

Significance. If the reported correlation holds, the metric offers a concrete, non-circular tool for monitoring VQE progress on near-term hardware where true ground states are inaccessible. The work is credited for explicit inclusion of energy-plateau cases and hardware runs on noisy trapped-ion devices for both 1D and 2D instances, directly addressing a common practical challenge in VQE.

major comments (1)
  1. [Abstract and results] Abstract and results summary: the central claim of a 'useful correlation' with fidelity and a 'helpful indication' of ground-state proximity is stated without accompanying quantitative measures (e.g., correlation coefficients, R² values, or error analysis across VQE iterations) for the 6- and 11-qubit instances; this leaves the strength of evidence for the primary claim only moderately supported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and results] Abstract and results summary: the central claim of a 'useful correlation' with fidelity and a 'helpful indication' of ground-state proximity is stated without accompanying quantitative measures (e.g., correlation coefficients, R² values, or error analysis across VQE iterations) for the 6- and 11-qubit instances; this leaves the strength of evidence for the primary claim only moderately supported.

    Authors: We agree that quantitative measures would strengthen the evidence. The manuscript presents the correlation primarily through figures showing the distance and fidelity versus VQE iterations (including energy-plateau cases) for the 11-qubit 1D and 6-qubit 2D instances. In the revised manuscript we will add explicit Pearson correlation coefficients (with associated p-values) between Hamiltonian-reconstruction distance and ground-state fidelity, computed across iterations for both models. Where multiple independent runs are available we will also include basic error analysis or R² values from linear regression. These additions will appear in the results section and will be referenced in the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central claim rests on an independent Hamiltonian-reconstruction procedure applied to the VQE output state; the resulting distance is then compared empirically to fidelity on the tested Ising instances. No equation or result in the paper reduces the distance to the VQE energy, a fitted parameter, or a self-citation chain. The reported correlations and plateau-detection behavior are direct numerical/hardware observations rather than derivations that close on their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard quantum mechanics for states and operators plus the validity of the cited Hamiltonian-reconstruction method; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard postulates of quantum mechanics (Hilbert space, Hermitian operators as observables, eigenstate-eigenvalue relation)
    Invoked throughout to define ground states, fidelity, and reconstruction.

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