Thermodynamic sampling of disordered materials with an analog Hamiltonian Rydberg simulator

This post was contributed by Mao Lin, Bruno Camino, John Buckeridge and Scott M. Woodley

Many advanced materials — from battery electrodes to semiconductor alloys — owe their useful properties to atomic-scale disorder. But predicting how atoms arrange themselves at a given temperature is hard: the number of possible configurations explodes combinatorially and sampling them according to their thermodynamic weights can overwhelm classical computation.

In this post, we report results from our recent publication demonstrating how the QuEra Aquila device, available through Amazon Braket, can serve as a thermodynamic sampler for realistic material models [1]. Using quantum annealing, we map an energy model derived from classical density functional theory (DFT) onto the neutral atom quantum hardware and sample low-energy configurations of nitrogen-doped graphene. We validate the approach on a 28-site system via exhaustive enumeration, benchmark it on a 78-site system against classical Monte Carlo sampling and demonstrate temperature tuning through programmable atom spacing.

To determine the equilibrium dopant concentration as a function of temperature and chemical potential for nitrogen doped graphene, we consider graphene nanoflakes (a single layer of carbon atoms arranged in a honeycomb lattice) where each lattice site can be occupied by either a carbon or a nitrogen atom. Pure graphene, Figure 1(a), is taken as a reference state, and the energy of doped configurations is measured relative to this reference. Importantly, dopants do not contribute independently, and they interact with one another such that the energy depends not only on how many dopants are present but also on their relative arrangement on the lattice. In particular, the interaction strength decays with dopant separation so that the farther apart two dopants are, the weaker their mutual interaction. See Figure 1(b) for an illustration of the mapping and the interaction effects of the dopants.

Figure 1 – Illustration of mapping a doped graphene to Rydberg atoms. (a) A pristine graphene with 78 sites, each site can be occupied by either a carbon (gold) or a nitrogen atom (blue). (b) The energy levels of the doped graphene systems measured with respect to the pristine graphene (CM). Doping the graphene with a nitrogen atom (C(M-1)N1) increases the energy of the system by VDFT, the on-site energy of the dopant, whereas the energy of systems with two dopants (C(M-2)N2) depend also on the distance of the dopants. When the two dopants are next to each other, its energy relative to the pristine graphene is 2VDFT+VNNDFT where VNNDFT is the nearest neighbor interaction energy of the dopants. (c) The largest layout for the Rydberg atom simulator used this work which contains 78 sites. A valid layout must comply with the constraints that the nearest neighbor spacing is not less than RNNmin=4 μm and the y-coordinate difference between any two sites is not less than 2 μm.

We encoded the energy levels of a doped graphene using Rydberg atoms. In this setting, each atom acts as an effective two-level system with a ground state |g⟩ and a Rydberg excited state |r⟩, driven by a global laser field and coupled through distance-dependent Rydberg interactions. We chose to map carbon and nitrogen atoms to the ground and Rydberg states, respectively. A commonly used model for the resulting analog dynamics is the time-dependent Hamiltonian

Here ni is the Rydberg occupation operator, Ω(t) is the global Rabi frequency controlling coherent transitions between the ground and Rydberg states, Δ(t) is the global detuning that biases Rydberg excitation, and C6 is a constant that determines the van der Waals interaction between atoms i and j set by the programmable inter-atom spacing Ri,j. The cost function Hcost is a time-dependent function that depends on the global detuning, atom arrangement and the distribution of the Rydberg occupations.

At the end of the quantum annealing evolution, we want the Rydberg Hamiltonian to reproduce the same relative energy ordering and thermodynamic weights as the classical DFT formation-energy landscape of the doped graphene. To accomplish this we construct a training set of Ntrain doped graphene configurations represented as binary occupation vectors (carbon = 0 and nitrogen = 1), calculate the corresponding DFT formation energies, then solve the linear regression:

The resulting parameters, VDFT and Ri,jDFT, are used to parameterize the final time Hcost and the subsequent quantum annealing procedure on the Rydberg hardware.

The overall energy scale of the DFT-derived model is not commensurate with the scale accessible in the Rydberg device. Specifically, after solving the linear regression, the resulting parameters are orders of magnitude larger than what the device can realize given its bounded detuning range and the maximum van der Waals interaction set by the minimum allowed spacing RNNmin between the nearest-neighbor atoms. In other words, the nearest-neighbor spacing obtained by DFT, RNNDFT, is much smaller than RNNmin.

We address this mismatch of energy scales by defining a single uniform rescaling factor αV=(RNNmin/RNNDFT)6 that compresses the entire DFT energy model into the device's dynamic range. The largest target coupling is realized at RNNmin while preserving relative energy differences and thus the Boltzmann weights up to a temperature rescaling. The device therefore samples the original graphene thermodynamics at an effective temperature T'=αVTeff where Teff is a parameter to be calibrated. Moreover, because the nearest-neighbor spacing RNN is a programmable hardware choice (subject to RNN ≥ RNNmin), adjusting RNN provides a means to tune the effective temperature of the sampling, and hence study the temperature dependence of the dopant concentration.

In this work, we use an analog Hamiltonian Rydberg simulator as a quantum-annealer, which is often described as a ground-state finder. But this is the ideal limit of a perfectly isolated, zero-temperature, infinitely slow evolution. In practice, an analog Rydberg simulator runs for finite time and is weakly coupled to the environment, so its output is better viewed as a heuristic sampler that preferentially returns low-energy configurations rather than a deterministic optimizer. This is exactly what thermodynamic sampling requires: at temperature T, equilibrium assigns each configuration z, say a dopant pattern, probability P(z, T) ∝ exp(-E(z)/kBT), so the goal is not to always return the configuration that minimizes E(z), but to return low-energy configurations with higher frequency according to their Boltzmann weight. Empirically, many annealing devices produce output distributions that are often reasonably approximated by P(z, Teff) for some effective temperature Teff that can be calibrated [3-6]. In this perspective, the same non-idealities that limit guaranteed ground-state preparation — finite temperature, control noise, and incomplete adiabaticity — become a feature: they provide the mixing needed to sample an ensemble. For disordered materials, where thermodynamic observables are dominated by a relatively small set of low-energy configurations, a device that repeatedly produces Boltzmann-biased samples can be directly useful even without perfect equilibration over the entire configuration space.

We report three experiments: effective-temperature calibration on a 28-site system, a 78-site benchmark against unbiased Monte Carlo, and temperature tuning through the programmable nearest-neighbor spacing.

Calibrating the effective temperature. We first validate the mapping and calibrate the sampler's effective temperature on a small instance where exhaustive search is possible. Using a 28-site graphene nanoflake mapped to a Rydberg-atom array with nearest-neighbor spacing fixed to the hardware minimum, we run the QPU