Thermodynamic sampling of materials using neutral-atom quantum computers

Technical Summary: Thermodynamic Sampling of Materials Using Neutral-Atom Quantum Computers

Problem Statement
Technologically relevant materials often exhibit configurational disorder (e.g., solid solutions, vacancies, substitutional alloys). Predicting their properties requires not only accurate energetic models but also methods capable of sampling the vast space of atomic configurations to determine equilibrium states at specific temperatures and chemical potentials. Classical approaches, such as Monte Carlo sampling and simulated annealing, face computational bottlenecks as system sizes increase, rendering exhaustive enumeration intractable. While quantum annealing has been explored for this purpose, mapping realistic material models—specifically those derived from Density Functional Theory (DFT)—onto current quantum hardware presents significant challenges. These include hardware constraints on interaction ranges, energy scales, and the lack of native support for periodic boundary conditions in neutral-atom arrays. Furthermore, establishing a rigorous thermodynamic interpretation for the output distributions of quantum optimizers remains an open challenge, as many devices are better described by coherent unitary dynamics than equilibrium thermal ensembles.

Methodology
The authors propose a framework to map the thermodynamic properties of nitrogen-doped graphene onto a Rydberg-atom Hamiltonian suitable for neutral-atom quantum hardware (specifically the QuEra Aquila device). The methodology proceeds through the following steps:

1. DFT Data Generation: Formation energies for nitrogen-doped graphene are calculated using periodic DFT (CRYSTAL23 code) for a 78-atom supercell. Symmetry-independent configurations (SICs) are generated and relaxed; symmetry-equivalent configurations (SECs) are derived to expand the training and test sets.
2. Hamiltonian Mapping: The DFT formation energies are fitted to a Rydberg Hamiltonian. This involves minimizing the error between DFT energies and a model comprising an on-site term ($V$) and distance-dependent pair interactions ($C_6/R^6$). The fit yields an on-site energy $V^{DFT}$ and an effective nearest-neighbor distance $R^{DFT}_{NN}$.
3. Hardware Rescaling Strategy: A critical mismatch exists between the DFT-derived parameters and hardware capabilities. The hardware requires a minimum interatomic spacing ($R^{min}_{NN} = 4\,\mu\text{m}$), whereas the fitted $R^{DFT}_{NN}$ is significantly smaller ($\approx 1.6\,\mu\text{m}$). Consequently, the native two-body interaction strength on the hardware is orders of magnitude weaker than the target material interaction.
   • To resolve this, the authors introduce a rescaling factor $\alpha_v = (R^{min}_{NN} / R^{DFT}_{NN})^6 \approx 236.69$.
   • Both the on-site energy and the pair interaction are scaled by $\alpha_v$ to fit within the hardware's accessible energy window.
   • This rescaling establishes a correspondence between the hardware sampling temperature $T$ and an effective material temperature $T' = \alpha_v T$.
4. Chemical Potential Control: The global laser detuning ($\Delta_g$) in the quantum annealer is mapped to the grand-canonical chemical potential difference ($\Delta\mu$) of the material. The rescaling factor allows the hardware's limited detuning range to cover a chemically relevant range of $\Delta\mu$ in the material model.
5. Validation:
   • 28-site System: The mapping is validated against an exhaustive classical enumeration of all possible configurations.
   • 78-site System: For the larger system, where exhaustive enumeration is impossible, results are benchmarked against random Monte Carlo (MC) sampling.
   • Temperature Tuning: The effective temperature is experimentally tuned by varying the nearest-neighbor spacing ($R_{NN}$) in the hardware, demonstrating control over the sampled distribution.

Key Contributions

• Rescaling Framework: The paper introduces a systematic rescaling procedure ($\alpha_v$) that bridges the gap between DFT-derived energy scales and the limited dynamic range of current neutral-atom hardware. This allows the hardware to sample distributions corresponding to the target material at an effective temperature $T'$.
• Thermodynamic Interpretation: The work provides a rigorous derivation linking the hardware control parameters (laser detuning and interatomic spacing) to thermodynamic variables (chemical potential and temperature), enabling the interpretation of quantum annealing outputs as Boltzmann-like distributions.
• Hardware Constraints Handling: The methodology explicitly addresses geometric constraints (minimum atom spacing, 2D arrangement) and the lack of periodic boundary conditions in neutral-atom devices, treating finite nanoflakes as representative of bulk behavior.
• Proof of Concept: The study successfully demonstrates the extraction of thermodynamic properties (average composition and energy distributions) for a doped material system using a programmable quantum annealer.

Results

• 28-Site Benchmark: For the 28-site graphene nanoflake, the quantum annealer's output (average nitrogen concentration vs. chemical potential) closely matches the exact results from exhaustive enumeration. The best fit corresponds to an effective sampling temperature of $T = 41\,\mu\text{K}$ (yielding a material effective temperature of $T' \approx 9.7\,\text{mK}$). The Total Variation Distance (TVD) between the quantum and classical distributions is 0.294, indicating moderate agreement, with deviations attributed to the sensitivity of low-energy states to temperature.
• 78-Site Benchmark: For the 78-site system, the quantum annealer preferentially samples low-energy configurations, whereas random Monte Carlo sampling explores the full energy landscape uniformly. The quantum device successfully identifies configurations with zero energy (non-interacting nitrogen arrangements) that are combinatorially rare and difficult to capture via random sampling.
• Temperature Control: By increasing the interatomic spacing from $4\,\mu\text{m}$ to $5\,\mu\text{m}$, the authors successfully demonstrated the ability to tune the effective sampling temperature from $\approx 9.7\,\text{mK}$ to $\approx 36\,\text{mK}$, confirming the thermodynamic consistency of the scaling strategy.

Significance and Claims
The authors position this work as a proof-of-concept demonstration of mapping realistic material models (derived from DFT) to Rydberg Hamiltonians for quantum annealing.

• The primary significance lies in establishing a workflow that enables the implementation of realistic material models on available quantum hardware despite severe energy-scale mismatches.
• The paper claims that the proposed rescaling strategy allows the hardware to sample distributions that can be interpreted in thermodynamic terms, specifically as Boltzmann-like ensembles at an effective temperature.
• The authors acknowledge current limitations, noting that the method is restricted to two-body interactions and 2D systems due to hardware constraints. They explicitly state that systems with strong localized interactions or many-body effects (e.g., phosphorous-nitrogen co-doping) cannot be accurately captured by the current two-body Hamiltonian and would require higher-order couplings or more flexible interaction models.
• The work does not claim to outperform classical methods for the specific systems studied but rather demonstrates the feasibility of using neutral-atom quantum computers as a component in future hybrid workflows for materials discovery.