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

This paper reports the first successful application of a modified Quantum Annealer Eigensolver on D-Wave hardware to calculate magnetic dipole hyperfine structure constants for neutral and Li/Na-like atoms, demonstrating results consistent with high-precision classical GRASP calculations within a three-decimal-place precision limit.

Original authors: Boni Paul (Centre for Quantum Engineering, Research and Education, Department of Physical Sciences, Indian Institute of Technology Tirupati, Andhra Pradesh, India), Subimal Deb (Centre for Quantum Eng

Published 2026-06-19

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CC BY 4.0

Original authors: Boni Paul (Centre for Quantum Engineering, Research and Education, Department of Physical Sciences, Indian Institute of Technology Tirupati, Andhra Pradesh, India), Subimal Deb (Centre for Quantum Engineering, Research and Education), Per Jönsson (Department of Materials Science and Applied Mathematics, Malmö University, Malmö, Sweden), Jörgen Ekman (Department of Materials Science and Applied Mathematics, Malmö University, Malmö, Sweden), Bhanu Pratap Das (Centre for Quantum Engineering, Research and Education)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the absolute lowest point in a vast, foggy mountain range. In the world of physics, this "lowest point" represents the most stable, calm state of an atom (its ground state). Usually, scientists use powerful classical supercomputers to map this terrain. But this paper reports a new experiment: using a special type of "quantum mountain climber" called a Quantum Annealer to find these low points and measure something very specific about the atom.

Here is a simple breakdown of what the researchers did, using everyday analogies:

1. The Goal: Measuring the Atom's "Heartbeat"

Atoms aren't just empty space; they have a nucleus (the center) and electrons (the dancers) swirling around it. The nucleus has a tiny magnetic field, like a microscopic bar magnet. The electrons also have their own magnetic fields. When these two magnets interact, it creates a subtle "hum" or vibration in the atom's energy levels. Scientists call this the Hyperfine Structure.

Think of it like a guitar string. If you pluck it, it makes a note. But if you slightly change the tension or the thickness of the string (like the interaction between the nucleus and electron), the pitch changes just a tiny bit. The researchers wanted to calculate exactly how much that pitch changes. This is crucial because these tiny changes are used in the world's most precise clocks (atomic clocks).

2. The Tool: The Quantum Annealer

To solve the math behind this, the team used a D-Wave Quantum Annealer.

The Analogy: Imagine you have a giant, complex maze. A classical computer is like a person who walks every single path one by one to find the exit. A Quantum Annealer is like a magical ghost that can "tunnel" through walls and explore many paths at once using quantum magic (superposition and tunneling) to find the exit much faster.
The Algorithm: They used a specific recipe called QAE (Quantum Annealer Eigensolver). Think of this as a specialized map-reading tool that tells the quantum machine exactly how to navigate the maze to find the "ground state" (the lowest energy point).

3. The Challenge: Too Many Variables

The math for these atoms is incredibly complex. It involves thousands of possible ways the electrons can arrange themselves (called Configuration State Functions or CSFs).

The Problem: The quantum machine they used is like a small, early-model smartphone. It doesn't have enough "memory" or "processing power" to handle the full, massive map of the atom.
The Solution (The "Zoom" Trick): The researchers invented a clever workaround. Instead of trying to load the whole map at once, they used a "Zoom-and-Sigma" strategy.
- Zooming: They started with a rough, wide-angle view of the problem to get a general idea of where the answer lies. Then, they "zoomed in" closer and closer, refining their guess step-by-step.
- Truncation: They realized that in the specific atoms they studied (Lithium, Beryllium, Sodium, Magnesium), only a few electron arrangements really mattered. They cut out the "noise" (the unimportant paths) and kept only the top 10-12 most important ones. This made the problem small enough for the quantum machine to handle.

4. The Experiment: Testing the Machine

They tested this method on four different atoms:

Neutral Lithium (Li)
Lithium-like Beryllium (Be+)
Neutral Sodium (Na)
Sodium-like Magnesium (Mg+)

They compared the results from their Quantum Annealer against two other methods:

GRASP: The gold-standard classical supercomputer calculation (the "expert human").
Simulated Annealing: A classical computer method that mimics the quantum process but without the quantum magic.

5. The Results: A Perfect Match

The paper claims a major success:

Accuracy: The quantum machine's results matched the classical "expert human" (GRASP) results almost perfectly.
Precision: They were accurate to three decimal places. For example, if the magnetic "hum" was 285.938 MHz, the quantum computer calculated 285.938 MHz.
Consistency: Whether they looked at the energy of the atom or the magnetic "hum" (Hyperfine constant), the quantum machine got it right.

6. Key Takeaways

It Works: This is the first time anyone has successfully used a quantum annealer to calculate these specific magnetic properties (Hyperfine constants) for atoms, rather than just calculating simple energy levels.
The "S-Orbital" Secret: They found that for these light atoms, the "s-orbitals" (a specific shape of electron cloud that gets very close to the nucleus) are the most important players. Including these in their simplified model was the key to getting accurate results.
Future Potential: While the current machine had limits (it could only handle small maps), the success proves that quantum annealing is a viable tool for solving complex atomic problems. As the hardware gets better (like the upcoming "Advantage2" mentioned in the paper), they believe they can tackle even more complex atoms and properties.

In short: The researchers taught a quantum computer how to solve a very specific, difficult math puzzle about how atoms vibrate. By using a "zoom-in" strategy and simplifying the problem, the quantum computer solved it with the same accuracy as the best classical supercomputers, proving that quantum machines are ready to help us understand the tiny details of the atomic world.

Technical Summary: Applications of Quantum Annealing to Magnetic Dipole Hyperfine Structure Constants

Problem Statement
Accurate calculation of atomic properties beyond ground-state energies, such as magnetic dipole hyperfine structure (HFS) constants, presents significant challenges due to the need to capture subtle electron correlation effects, particularly in the near-nuclear region where relativistic and non-relativistic effects are pronounced. While quantum annealing (QA) has been applied to ground-state energy calculations and other eigenvalue problems, its application to HFS constants—which impose stricter requirements on wavefunction precision than energy determinations—remains unexplored. This work addresses the capability of quantum annealers to compute these constants for neutral Lithium (Li), Lithium-like Beryllium (Be $^+$ ), neutral Sodium (Na), and Sodium-like Magnesium (Mg $^+$ ).

Methodology
The authors employed a modified Quantum Annealer Eigensolver (QAE) algorithm on D-Wave's quantum hardware to solve the relativistic Dirac-Coulomb Hamiltonian ( $H_{DC}$ ) eigenvalue problem. The computational workflow integrates classical and quantum components:

Classical Preprocessing: Using the General-purpose Relativistic Atomic Structure Package (GRASP), the authors generated:
- Matrix elements $H_{rs} = \langle \Phi_r | H_{DC} | \Phi_s \rangle$ defining the electronic structure.
- Reduced matrix elements for the magnetic dipole hyperfine operator ( $T_e^{(1)}$ ).
- Configuration State Functions (CSFs) based on Multiconfiguration Dirac-Hartree-Fock (MCDHF) orbitals.
QAE Formulation: The eigenvalue problem was reformulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. The energy functional $\epsilon(\lambda) = \langle \Psi | H_{DC} | \Psi \rangle - \lambda \langle \Psi | \Psi \rangle$ was mapped to a QUBO form using a floating-point encoding scheme, where expansion coefficients $c_r$ were represented by $K$ -bit binary variables.
Modified QAE Strategy: To estimate the ground-state eigenvalue and eigenvector, the authors adopted a "zooming-and-sigma-annealing" approach:
- Zooming: An iterative process starting with a coarse scan of the Lagrange multiplier $\lambda$ around the diagonal element $H_{11}$ , followed by progressively narrower windows centered on the interpolated root of the energy functional to refine $\lambda_{opt}$ .
- Sigma-annealing: An iterative convergence of the coefficients $\{c_r\}$ .
Basis Truncation: Due to hardware limitations (specifically the lack of all-to-all qubit connectivity and the constraint of 10 qubits for $K=10$ ), the CSF basis was truncated. For systems requiring more than 11 CSFs, a truncation scheme retained only the most significant CSFs (up to 12) based on initial estimates derived from the independent electron pair approximation ( $c_r \approx H_{r1}/(H_{11} - H_{rr})$ ).
Post-Processing: The quantum-derived coefficients $\{c_r\}$ were contracted with the classically computed operator matrix elements to extract the HFS constant $A$ using the Wigner-Eckart theorem.

Key Results
The study benchmarked the modified QAE results against classical GRASP calculations and Simulated Annealing (SA) for four atomic systems ( $^7$ Li, $^9$ Be $^+$ , $^{23}$ Na, $^{25}$ Mg $^+$ ) in their lowest angular symmetry states ( $J=1/2$ , even parity).

Accuracy: The quantum annealing results were completely consistent with GRASP and SA within the chosen precision (three decimal places).
- For Li and Be $^+$ , relative deviations in ground-state energies were below 0.03% and 0.2% respectively, while HFS constants showed 0.000% deviation across all methods.
- For Na and Mg $^+$ , energy deviations remained within 0.02%, and HFS constants matched GRASP/SA results with deviations generally below 0.01%.
Orbital Sensitivity: The results highlighted that s-orbital correlation dominates hyperfine interactions in light alkali-like systems. For heavier systems (Na, Mg $^+$ ), accurate results required the inclusion of virtual $p_{1/2}$ orbitals alongside s-orbitals to account for enhanced relativistic effects and nuclear charge distribution.
Coefficient Precision: While absolute errors in higher-order coefficients ( $c_{r>1}$ ) were larger, their impact on the final HFS constant was negligible due to the quadratic dependence of $A$ on the coefficients and the dominance of the ground-state coefficient $|c_1|$ .

Significance and Claims
The paper claims to present the first results of magnetic dipole HFS constants calculated using a quantum annealer, extending the QAE framework beyond previously studied ground-state atomic energies.

Validation: The work demonstrates that quantum annealers, when coupled with a modified algorithm featuring zooming and sigma-annealing, can accurately capture the subtle many-body effects required for HFS calculations, yielding results consistent with state-of-the-art classical relativistic methods.
Scalability: The study establishes a proof-of-concept for calculating atomic properties on current quantum hardware (D-Wave Advantage system 6.4) by effectively managing qubit constraints through basis truncation and floating-point encoding.
Future Outlook: The authors suggest that the proposed framework can be readily extended to other atomic properties, specifically electric-quadrupole HFS constants and isotope-shift parameters. They note that future hardware advancements (e.g., D-Wave Advantage2 with Zephyr topology) and improved solvers are expected to further reduce embedding overheads and enhance solution accuracy, potentially enabling quantum annealers to play a more significant role in determining atomic and molecular properties.