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arxiv: 2606.20166 · v2 · pith:ELIG3PQTnew · submitted 2026-06-18 · 🪐 quant-ph · physics.atom-ph· physics.chem-ph

Applications of quantum annealing to magnetic dipole hyperfine structure constants: First results beyond energies for atoms

Boni Paul , Subimal Deb , Per J\"onsson , J\"orgen Ekman , Bhanu Pratap Das This is my paper

Pith reviewed 2026-06-30 10:39 UTC · model grok-4.3

classification 🪐 quant-ph physics.atom-phphysics.chem-ph
keywords quantum annealinghyperfine structureD-Wavemagnetic dipoleatomic structureconfiguration state functionsGRASPDirac-Coulomb Hamiltonian
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The pith

Quantum annealing on D-Wave hardware computes magnetic dipole hyperfine structure constants for Li, Be, Na, and Mg atoms to three decimal places, matching GRASP relativistic calculations.

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

The paper applies a modified Quantum Annealer Eigensolver to the problem of magnetic dipole hyperfine structure constants in light atoms and their ions. Relativistic Dirac-Coulomb Hamiltonian matrices are built from at most twelve configuration state functions, and a zooming-and-sigma-annealing scheme with floating-point encoding extracts the ground-state eigenvector on D-Wave hardware. Hardware results are benchmarked directly against GRASP multiconfiguration Dirac-Hartree-Fock values and against simulated annealing, showing agreement at the reported precision. This constitutes the first reported use of quantum annealing for hyperfine constants rather than energies alone.

Core claim

The modified QAE algorithm, run on the D-Wave QPU, produces magnetic dipole HFS constants for neutral Li, Li-like Be, neutral Na, and Na-like Mg that remain consistent with full GRASP calculations when the H_DC matrix is limited to eleven or fewer CSFs or to a truncated set of twelve significant CSFs, with all values agreeing to three decimal places.

What carries the argument

Modified Quantum Annealer Eigensolver (QAE) with zooming-and-sigma-annealing and floating-point encoding, applied to small relativistic Dirac-Coulomb Hamiltonian matrices constructed from truncated sets of configuration state functions.

If this is right

  • Hardware QPU output for the four atomic systems agrees with GRASP at three decimal places for the chosen matrix sizes.
  • The accuracy achieved depends on the specific atom and on the dimension of the H_DC matrix.
  • The CSF truncation scheme permits use of extended correlation orbital sets while preserving the reported precision.
  • Quantum annealing supplies an alternative route to hyperfine constants once the ground-state eigenvector is obtained.

Where Pith is reading between the lines

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

  • Scaling the same encoding to matrices larger than twelve CSFs would require more qubits or improved precision schemes.
  • The method could be tested on electric quadrupole or magnetic octupole constants using the same Hamiltonian matrices.
  • If the truncation error remains small for heavier atoms, the approach might serve as a hybrid quantum-classical tool for selected atomic properties.

Load-bearing premise

Retaining only the twelve configuration state functions that contribute most to the ground-state wavefunction keeps the magnetic dipole HFS constant accurate to three decimal places.

What would settle it

A GRASP run that includes the full untruncated CSF basis for any of the four systems and produces an HFS constant differing by more than 0.001 from the QAE result would falsify the accuracy claim.

Figures

Figures reproduced from arXiv: 2606.20166 by Bhanu Pratap Das, Boni Paul, J\"orgen Ekman, Per J\"onsson, Subimal Deb.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The initial guess for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The absolute error in coefficients computed by SA [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1: The absolute error in coefficients computed by SA and QA with respect to GRASP for Li for the cases [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The absolute error in coefficients computed by SA and QA with respect to GRASP for Na for the cases [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The absolute error in coefficients computed by SA and QA with respect to GRASP for Mg [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
read the original abstract

We report the first results of the magnetic dipole hyperfine structure (HFS) constants of neutral $\mathrm{Li}$, Li-like $\mathrm{Be}$, neutral $\mathrm{Na}$, and Na-like $\mathrm{Mg}$ using a modified version of the Quantum Annealer Eigensolver (QAE) algorithm on D-Wave's quantum hardware. The results are benchmarked against relativistic configuration interaction with multiconfiguration Dirac Hartree-Fock (MCDHF) calculations using the General-purpose Relativistic Atomic Structure Package (GRASP), and simulated annealing. In our modified QAE, a zooming-and-sigma-annealing approach with a floating-point encoding scheme is adopted to estimate the ground-state eigenvalue and eigenvector of the relativistic Dirac-Coulomb Hamiltonian matrices ($H_{\mathrm{DC}}$) constructed from 11 or fewer configuration state functions (CSFs). For calculations with extended correlation orbital sets, we applied a CSF truncation scheme, retaining only CSFs (up to 12) that make significant contributions to the ground-state wavefunction. Our modified QAE precision is kept limited to three decimal places (up to 10 qubits). Hardware demonstrations on the D-Wave quantum processing unit (QPU) yielded results that were completely consistent with GRASP (at the chosen precision) in determining the magnetic dipole HFS constants, with accuracy varying across systems and $H_{\mathrm{DC}}$ matrix dimensions.

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 / 1 minor

Summary. The manuscript reports the first application of a modified Quantum Annealer Eigensolver (QAE) on D-Wave hardware to compute magnetic dipole hyperfine structure (HFS) constants for neutral Li, Li-like Be, neutral Na, and Na-like Mg. It employs a zooming-and-sigma-annealing approach with floating-point encoding on Dirac-Coulomb Hamiltonian matrices constructed from at most 12 configuration state functions (CSFs), applies an energy-based CSF truncation for larger orbital sets, and claims that hardware results are fully consistent with GRASP relativistic CI calculations at the chosen three-decimal-place precision.

Significance. If the central claims hold, the work provides an initial hardware demonstration that quantum annealing can be extended from energy eigenvalues to a distinct one-body observable (magnetic dipole HFS) in relativistic atomic structure. The explicit benchmarking against an independent classical code (GRASP) and the use of actual QPU runs are positive features. However, the restriction to very small truncated bases and three-decimal precision limits the immediate impact on atomic physics.

major comments (2)
  1. [Method (CSF truncation scheme)] Method section on CSF truncation: the scheme retains at most 12 CSFs chosen by their contribution to the ground-state wavefunction of the Dirac-Coulomb Hamiltonian, yet no direct comparison (full vs. truncated basis) is shown to confirm that the truncation error on the magnetic dipole HFS constant remains below 0.001. Because the HFS operator has different radial weighting (especially near the nucleus) from the energy functional, energy-based selection does not automatically guarantee the reported precision for the HFS observable.
  2. [Results and abstract] Results and abstract: the floating-point encoding scheme and the precise mapping of eigenvector components to the HFS expectation value are only sketched; without an explicit error analysis or reported uncertainty for each system and matrix dimension, it is difficult to assess whether the claimed three-decimal consistency with GRASP is robust or an artifact of the limited precision (up to 10 qubits).
minor comments (1)
  1. [Abstract] Abstract: the statement that 'accuracy varying across systems and H_DC matrix dimensions' is not accompanied by any tabulated values or quantitative measure of that variation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below.

read point-by-point responses

  1. Referee: Method section on CSF truncation scheme: the scheme retains at most 12 CSFs chosen by their contribution to the ground-state wavefunction of the Dirac-Coulomb Hamiltonian, yet no direct comparison (full vs. truncated basis) is shown to confirm that the truncation error on the magnetic dipole HFS constant remains below 0.001. Because the HFS operator has different radial weighting (especially near the nucleus) from the energy functional, energy-based selection does not automatically guarantee the reported precision for the HFS observable.

    Authors: We agree the truncation criterion is energy-based and that a direct full-vs-truncated comparison for the HFS constant is not shown. Given the deliberately small bases (≤12 CSFs) used for this hardware demonstration, the dominant contributions are retained, but we acknowledge the different radial weighting of the HFS operator. In revision we will add an explicit statement noting this limitation and that quantitative truncation-error checks for the observable are left to future work with larger resources. revision: partial

  2. Referee: Results and abstract: the floating-point encoding scheme and the precise mapping of eigenvector components to the HFS expectation value are only sketched; without an explicit error analysis or reported uncertainty for each system and matrix dimension, it is difficult to assess whether the claimed three-decimal consistency with GRASP is robust or an artifact of the limited precision (up to 10 qubits).

    Authors: The floating-point encoding and the mapping of the extracted eigenvector to the HFS expectation value are described in the Methods. The reported consistency is at the three-decimal precision level matching our QAE implementation. We will expand the Results section with a clearer step-by-step description of the mapping together with per-system uncertainty estimates tied to the qubit precision and annealing parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external benchmark against independent GRASP code

full rationale

The paper solves the Dirac-Coulomb eigenvalue problem on D-Wave hardware via modified QAE, extracts the ground-state vector, and evaluates the magnetic dipole HFS operator as an expectation value on that vector. Results are compared directly to independent MCDHF calculations performed with the external GRASP package. The CSF truncation (retaining up to 12 functions by ground-state energy contribution) is an approximation whose effect on the distinct HFS observable is checked by the external benchmark rather than being forced by definition or by a self-citation chain. No load-bearing step reduces to a fitted parameter, a self-referential uniqueness theorem, or an ansatz imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The truncation threshold for CSFs and the three-decimal precision target function as implicit modeling choices whose impact on the HFS constant is not quantified.

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