Quantum-computing study of the electronic structure of crystals - the case study of Si
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2023-01-01 |
| Journal | NANOCOM ⌠|
| Authors | Michal ÄuriĹĄka, Ivana MihĂĄlikovĂĄ, Martin FriĂĄk |
| Institutions | Czech Academy of Sciences, Institute of Physics of Materials, Masaryk University |
| Citations | 1 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive SummaryâThis study validates the application of hybrid quantum computing techniques for calculating the electronic band structure of crystalline materials, using Silicon (Si) as a case study.
- Core Value Proposition: Demonstrates a viable methodology (VQE combined with tight-binding) for simulating solid-state physics problems on current, resource-limited quantum processors.
- Material System: Crystalline diamond-structure Silicon (Si), modeled using a simplified tight-binding Hamiltonian focusing only on s orbitals (requiring only one qubit).
- Methodology: Utilized the hybrid Variational Quantum Eigensolver (VQE) algorithm, employing the COBYLA classical optimizer for iterative minimization.
- Key Achievement (Accuracy): Achieved excellent agreement between quantum-simulated results and classical calculations for the two lowest occupied energy bands of Si.
- Computational Sensitivity: Critically assessed the sensitivity of results to the computational setup, finding that the maximum number of iterations (MAXITER) is the primary factor determining accuracy.
- Resource Optimization: Determined that lower-energy states converge reliably with fewer iterations than higher-energy states, providing a pathway for optimizing quantum resource allocation.
- Hardware Validation: Successfully obtained results for the lowest energy band using a real physical quantum processor provided by the IBM Quantum platform (âibm_lagosâ).
Technical Specifications
Section titled âTechnical Specificationsâ| Parameter | Value | Unit | Context |
|---|---|---|---|
| Material Studied | Silicon (Si) | N/A | Crystalline diamond structure. |
| Electronic Model | Tight-Binding (Slater-Koster) | N/A | Simplified model using only s orbitals. |
| On-Site Energy (Es) | -4.03 | eV | Parameterized value for Si s orbitals. |
| Hopping Energy (Vss) | -8.13 | eV | Parameterized value for Si s-s interaction. |
| Hamiltonian Size | 2x2 | N/A | Matrix size after s-orbital simplification. |
| Qubit Requirement | 1 | Qubit | Required for mapping the 2x2 Hamiltonian using Pauli matrices. |
| Quantum Algorithm | Variational Quantum Eigensolver (VQE) | N/A | Hybrid classical-quantum approach for finding ground energy. |
| Classical Optimizer | COBYLA | N/A | Constrained Optimization by Linear Approximation. |
| Minimum MAXITER (Simulator) | 20 | Iterations | Required for decent accuracy across both bands in simulation. |
| MAXITER (Real Hardware) | 100 | Iterations | Set value for reliable results on the IBM processor. |
| Quantum Processor Used | âibm_lagosâ | N/A | Real quantum hardware provided by IBM Quantum. |
Key Methodologies
Section titled âKey MethodologiesâThe study employed a hybrid quantum-classical approach based on the VQE algorithm to compute the eigenvalues (energies) of the Si Hamiltonian H(k) across the reciprocal space.
- Crystal Description: The electronic structure of diamond-structure Si was modeled using the tight-binding method (Slater and Koster).
- Hamiltonian Simplification: To minimize quantum resource usage, the standard multi-orbital model was simplified to include only interactions between s orbitals, resulting in a 2x2 Hamiltonian matrix H(k).
- Qubit Mapping: The 2x2 Hermitian Hamiltonian H(k) was transformed into the qubit space by decomposition using the complete Pauli basis (I, X, Y, Z), requiring only one qubit.
- VQE Cycle Initiation: A parametrized quantum state was prepared, and a probe point in the sample space (reciprocal space vector k) was selected.
- State Preparation (Ansatz): The heuristic RyRz variational form, consisting of parameterized single-qubit rotations, was used to prepare the quantum state.
- Measurement and Energy Calculation: Expectation values of each Hamiltonian term were measured on the quantum device (simulator or real hardware). Post-rotations were used to switch non-diagonal Hamiltonian terms to a diagonal basis for measurement.
- Classical Optimization: The energy was calculated, and the classical COBYLA optimization method was used to adjust the parameters of the quantum state iteratively until the maximum number of iterations (MAXITER) was reached.
- Validation: The computed quantum energies (blue points) were compared against the exact analytical solution obtained by classical diagonalization of the 2x2 tight-binding matrix (red curves).
Commercial Applications
Section titled âCommercial ApplicationsâThis research provides foundational validation for using quantum computation in materials science, which has implications for future semiconductor and computational industries.
- Advanced Semiconductor R&D: Enables rapid, potentially exponential speed-up in calculating the electronic properties (band gaps, effective masses) of complex crystalline materials, crucial for designing next-generation Si-based microprocessors and compound semiconductors.
- Quantum Algorithm Benchmarking: Provides a robust, low-qubit benchmark problem for testing and optimizing new VQE variants and error mitigation techniques on noisy intermediate-scale quantum (NISQ) devices.
- Quantum Hardware Utilization: Establishes best-practice guidelines (e.g., required MAXITER values) for efficiently utilizing limited and expensive real quantum computing resources (like IBM Quantum platforms).
- Materials Database Generation: Facilitates the creation of large, accurate databases of material properties calculated via quantum methods, accelerating machine learning applications in materials discovery.
- Solid-State Qubit Development: The methodology can be extended to simulate defects and impurities in Si, which is essential for understanding and engineering solid-state quantum bits (qubits).
View Original Abstract
Quantum computing is newly emerging information-processing technology which is foreseen to be exponentially faster than classical supercomputers.Current quantum processors are nevertheless very limited in their availability and performance and many important software tools for them do not exist yet.Therefore, various systems are studied by simulating the run of quantum computers.Building upon our previous experience with quantum computing of small molecular systems (see I. MihĂĄlikovĂĄ et al., Molecules 27 (2022) 597, and I. MihĂĄlikovĂĄ et al., Nanomaterials 2022, 12, 243), we have recently focused on computing electronic structure of periodic crystalline materials.Being inspired by the work of Cerasoli et al. (Phys.Chem.Chem.Phys., 2020, 22, 21816), we have used hybrid variational quantum eigensolver (VQE) algorithm, which combined classical and quantum information processing.Employing tight-binding type of crystal description, we present our results for crystalline diamond-structure silicon.In particular, we focus on the states along the lowest occupied band within the electronic structure of Si and compare the results with values obtained by classical means.While we demonstrate an excellence agreement between classical and quantum-computed results in most of our calculations, we further critically check the sensitivity of our results with respect to computational set-up in our quantum-computing study.A few results were obtained also using quantum processors provided by the IBM.