Graphene Analysis Advances with Quantum Elastic Network Models and Coupled Oscillators

Understanding the relationship between a material’s atomic structure and its macroscopic properties remains a significant challenge in materials science. Ioannis Kolotouros, Adithya Sireesh, and Stuart Ferguson, alongside colleagues from the University of Edinburgh and Q-CTRL, have developed a novel approach to overcome computational limitations in modelling these complex systems. Their research introduces Quantum Elastic Network Models (QENMs), leveraging a recently published algorithm to achieve an exponential advantage in simulating molecular vibrations. This work demonstrates the efficient simulation of planar materials, exemplified through a detailed analysis of a two-dimensional graphene sheet and its behaviour regarding heat transfer and structural rippling. The team estimates that simulating a centimetre-scale graphene sheet, a task currently requiring immense classical computing resources, could be achieved using a surprisingly small number of quantum bits.

This work explores the development and application of QENMs to graphene, a two-dimensional material with exceptional properties, constructing a harmonic potential based on the Hessian matrix derived from density functional theory calculations and subsequently quantising the resulting vibrational modes. The resulting quantum Hamiltonian is then diagonalised to obtain energy levels and eigenstates, providing insights into the material’s vibrational properties and thermal behaviour.

Calculations were performed on graphene sheets of varying sizes to assess the model’s convergence and accuracy, with comparisons made to established classical methods. Results demonstrate that QENMs can accurately reproduce the low-frequency vibrational modes of graphene, crucial for understanding its thermal conductivity and mechanical properties, and investigate the influence of isotopic substitution on the vibrational density of states. The efficiency of the QENM approach allows for the investigation of larger graphene systems than are typically feasible with traditional molecular dynamics, opening avenues for studying defects and edge effects. in 2023. This approach leverages the potential for exponential speedup when simulating systems of coupled oscillators, enabling the efficient simulation of planar materials previously beyond reach. The team meticulously analysed the computational complexity associated with initial state preparation, Hamiltonian simulation, and measurement within the QENM framework applied to a two-dimensional graphene sheet.

This analysis revealed that simulating a centimeter-scale graphene sheet, a task requiring approximately 180 petabytes of memory using classical methods, could potentially be achieved with as few as 160 logical qubits, delivering a pathway to overcome the memory limitations of classical simulations. Scientists developed a detailed complexity analysis, quantifying the resources needed for each stage of the quantum simulation process, including the number of quantum gates and circuit depth required to accurately represent the graphene sheet’s atomic interactions. This work addresses the limitations of classical hardware in modelling materials at the atomic level across macroscopic scales, even when employing simplified elastic network models. The team successfully demonstrated the efficient simulation of planar materials, exemplified by a detailed analysis of a two-dimensional graphene sheet, revealing that an atomistic simulation of a graphene sheet spanning one centimetre squared could potentially be encoded using only 150-200 logical qubits. The research details a method for loading samples of the Maxwell-Boltzmann distribution onto a quantum state, requiring minimal resources.

Crucially, the study confirms that, under specific conditions , sparsity in connectivity and limited initial conditions , the algorithm introduced by Babbush et al. can be efficiently applied, constructing an efficient connectivity oracle for a graphene sheet and demonstrating sustained efficiency. Measurements confirm the algorithm’s ability to estimate global quantities like kinetic and potential energy of subsystems within the simulated material. Tests prove the feasibility of simulating a 1cm2 square graphene sheet, a task that would classically demand hundreds of petabytes of memory and prohibitive computational time.

The team meticulously analysed the complexity of initial state preparation, Hamiltonian construction, and measurement processes for this material, demonstrating the potential for simulating complex phenomena such as heat transfer and out-of-plane rippling effects within the graphene sheet. Furthermore, the study introduces a discretization method and establishes that all oracles required for the Babbush et al. algorithm can be efficiently executed in certain well-structured molecules. Building upon a previously established framework for simulating coupled oscillators, the authors demonstrate how quantum elastic network models can be constructed and efficiently mapped onto a quantum mechanical system, detailing methods for preparing initial states, representing molecular dynamics, and measuring relevant observables, specifically within the context of a two-dimensional graphene sheet. The research establishes that simulating an atomistic graphene sheet on the centimeter scale, a task requiring substantial classical computational resources, could potentially be achieved using a significantly reduced number of logical qubits. Importantly, the authors acknowledge that realizing a full exponential advantage remains challenging, particularly as system complexity increases through factors like doping or defects, and that further investigation is required. Future research should focus on extending the methodology to more complex materials and assessing the scalability of the approach with increasing system size and intricacy.