![Outline
2
• Motivation [3]
• Background
- NIST-JARVIS infrastructure [4-5]
- Band theory of solids [6-7]
- Quantum algorithms and circuits [8-13]
• Workflow development [14-20]
• Application to Aluminum metal [37-46]
• Application to more than 1000 solids [21-22]
• Application for many-body methods [ 23 ]
• Hands-on session [24]
• Summary and future work [25]
Contents Slide #](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Quantum Computation for Predicting Electron and Phonon Properties of Solids
- 1. Quantum Computation for Predicting Electron and Phonon Properties of Solids Kamal Choudhary NIST, Gaithersburg, MD, USA & Theiss Research, CA, USA. Developer & Founder: https://jarvis.nist.gov Washington DC Quantum Computing Meetup, 02/26/2022 1 Joint Automated Repository for Various Integrated Simulations
- 2. Outline 2 • Motivation [3] • Background - NIST-JARVIS infrastructure [4-5] - Band theory of solids [6-7] - Quantum algorithms and circuits [8-13] • Workflow development [14-20] • Application to Aluminum metal [37-46] • Application to more than 1000 solids [21-22] • Application for many-body methods [ 23 ] • Hands-on session [24] • Summary and future work [25] Contents Slide #
- 3. Motivation 3 Simulation of quantum systems might be easier using quantum bits (qubits) Energy levels (eigenvalues) of a Hamiltonian (Hermitian matrix) Determine metal/semiconductor/insulator Importance of Wannier tight-binding approach Universal strategy for electron and phonon dynamics of solids Analyzing role of different circuit models & optimizers Storing large Hamiltonian matrix (2nx2n, n:qubits)
- 4. Background: NIST-JARVIS Infrastructure 4 https://jarvis.nist.gov “You guys are doing something really beneficial…” “I find JARVIS-DFT very useful for my research…” User-comments: Established: January 2017 (MGI funded) Published: >25 articles Users: >8000 users worldwide Downloads: >300K Workshops: 2 AIMS, 1 QMMS (~200 attendees for each) jarvis.nist.gov: Requires login credentials, free registration Choudhary et al., Nature: npj Computational Materials 6, 173 (2020).
- 5. 2017 2018 2019 2020 2021 JARVIS-FF (Evaluate FF) JARVIS-DFT 2D (OptB88vdW, Exf. En.) JARVIS-DFT Optoelectronics (TBmBJ) JARVIS-DFT Elastic Tensor 3D & 2D JARVIS-ML CFID descriptors JARVIS-FF (Evaluate FF, defects) JARVIS-DFT Topological SOC spillage 3D JARVIS-DFT /ML K-point convergence JARVIS-DFT Solar SLME JARVIS-DFT Topological SOC spillage 2D (Mag/Non- Mag.) JARVIS-DFT/ML 2D Heterostructures JARVIS-DFT/ML DFPT Dielec., Piezo., IR JARVIS-DFT/ML Thermoelectrics 3D & 2D Seebeck, PF JARVIS-DFT EFG NQR, NMR JARVIS-DFT STM 2D JARVIS-DFT AQCE 2D JARVIS-DFT WTBH JARVIS-DFT/ML Topological SOC spillage 3D Mag., non-mag, Exp. JARVIS-AtomQC VQE/VQD JARVIS-DAC MOFs JARVIS- AtomVison (STEM/STM) JARVIS-TB TB3PY JARVIS- ALIGNN (Solids Molecules) JARVIS-ML UQ, Phonons JARVIS- OPTIMADE 5
- 6. Background: Band-theory of Solids 6 • Free electron model: electrons are weekly bound to their atoms and can move freely (Arnold Sommerfeld), especially applicable to metallic solids, do not take potential into account, several problems such as wrong description of specific heat for transition elements. • Nearly free electron model: electrons are ‘nearly’ free introducing weak lattice potential, wavefunctions (Bloch Wave) still represented by plane waves, takes potential into account. • Tight-binding model/LCAO: opposite extreme to NFE model, electrons are tightly bound to nuclei, atomic description is not completely irrelevant , Coulomb interactions between the atom cores and electrons split the energy levels to form bands, usually good for valence electrons, different methods/packages such as WTBH (Wannier90), TB3Py, DFTB etc. • Density functional theory: Instead of wavefunctions, consider density functional, many electron problem to many one-electron problem, effect of exchange-correlation functional. Many software such as VASP, PWSCF/QE, WIEN2k etc. • Bloch’s theorem: solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. • Brillouin zone: primitive cell in reciprocal space, set of points in k-space that can be reached from the origin without crossing any Bragg plane. • Band-structure: ranges of energy that an electron is "forbidden" or "allowed" to have, due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice. https://en.wikipedia.org/wiki/Brillouin_zone https://en.wikipedia.org/wiki/Electronic_band_structure
- 7. Background: Metal vs Insulator 7
- 8. Background: Feynman’s seminal papers 8 http://physics.whu.edu.cn/dfiles/wenjian/1_00_QIC_Feynman.pdf “Nature is quantum, goddamn it! So if we want to simulate it, we need a quantum computer.”
- 9. Background: NISQ 9 https://arxiv.org/abs/2101.08448 https://towardsdatascience.com/quantum-advantage-b3458646bd9 Companies: • Google, • Microsoft, • IBM • Xanadu, • D-Wave, • Rigetti…
- 10. Background: QPE & VQE 10 Two main approaches to ab-initio chemistry calculations on quantum computers: 1) Quantum phase estimation (QPE): estimate the phase (or eigenvalue) of an eigenvector of a unitary operator, U 2) Variation quantum eigensolver (VQE): a quantum/classical hybrid algorithm that can be used to find eigenvalues of a (often large) matrix, H
- 11. Variational Quantum Eigensolver (VQE) & Variation Quantum Deflation(VQD) 11 http://openqemist.1qbit.com/docs/vqe_microsoft_qsharp.html Notes: • Quantum computers are good in preparing states, not good at sum, optimizers, multiplying etc. • QC to prepare a wavefunction ansatz of the system and estimate the expectation value VQD: Deflate other eigensatets once ground state is found using VQE VQE: a hybrid classical-quantum algorithm using Ritz variational principle
- 12. VQE: Lack of application for solids 12
- 13. Background: Quantum Circuit Model 13 • In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly other actions. Any quantum program can be represented by a sequence of quantum circuits. Analogous to tuning parameters of a guitar, to an extent! https://qiskit.org/documentation/apidoc/circuit.html https://good-loops.com/beautiful-anime-guitar-strumming-gif/ https://qiskit.org/textbook/ch-algorithms/quantum-fourier-transform.html
- 14. Application to solid-state materials 14 https://github.com/usnistgov/jarvis https://github.com/usnistgov/atomqc arXiv:2102.11452
- 15. Typical Flowchart 15 https://github.com/usnistgov/jarvis https://github.com/usnistgov/atomqc K. Choudhary, J. Phys.: Condens. Matter 33 (2021) 385501 Wannier functions: • Complete orthonormalized basis set, • Acts as a bridge between a delocalized plane wave representation and a localized atomic orbital basis • All major density functional theory (DFT) codes support generation WFs for a material 𝐻 = ℎ𝑃𝑃 𝑃∈ 𝐼,𝑋,𝑌,𝑍 ⨂𝑛 𝐻𝑗 = 𝐻 + 𝛽𝑖|𝜓 𝜽0 ∗ 𝜓 𝜽0 ∗ | 𝑗−1 𝑖=0 𝐺 𝑘, ꞷ𝑛 = ꞷ𝑛 + 𝜇 − 𝐻 𝑘 − 𝛴 ꞷ𝑛 −1 http://www.wannier.org/
- 17. Circuit Trials 17 RealAmplitudes PauliTwoDesign EfficientSU2 jarvis.core.circuits.QuantumCircuitLibrary RY and RZ: parametrized circuits with parameters ө, Wires and boxes with ‘X’ :Controlled-X gate. Wires with two solid squares: Controlled-Z gates
- 18. Circuit Trials 18 Finding right circuit and number of repeat units are important a) Al for Gamma point, b) Al for X point, c) PbS for X point for different repeat units of Circuit-6
- 19. FCC Aluminum Example 19 a) Monitoring VQE optimization progress with several local optimizers such COBYLA, L_BFGS_B, SLSQP, CG, and SPSA for Al electronic WTBH and at X-point. b) Electronic bandstructure calculated from classical diagonalization (Numpy-based exact solution) and VQD algorithm for Al. c) Phonon bandstructure for Al
- 20. Application to ~1000 systems 20 Comparison of minimum (Min.) and maximum (Max.) energy levels at Г-point for electronic and phonon WTBH using classical eigenvalue routine in Numpy (Np.) and VQE solver. (N_qubits <=5) a) and b) comparison of phonon (Phn.) minimum and maximum energy levels for 930 materials, c) and d) comparison of electronic (El.) minimum and maximum energy levels for 300 materials. The colorbar represents the number of Wannier orbitals.
- 21. Application to ~1000 systems 21
- 22. Dynamical Mean Field Theory 22 Imaginary part of Al’s DMFT hybridization function for a few components considering zero self-energy. a)Δ00, b)Δ01, c)Δ10, d)Δ11 • Dynamical mean-field theory (DMFT): commonly used techniques for solving predicting electronic structure of correlated systems using impurity solver models. • DMFT maps a many-body lattice problem to a many- body local problem with impurity models. • In DMFT one of the central quantities of interest is the Green’s function such as 𝐺 𝑘, ꞷ𝑛 = ꞷ𝑛 + 𝜇 − 𝐻 𝑘 − 𝛴 ꞷ𝑛 −1 • Spectral function (𝐴) & DMFT hybridization function (𝛥) 𝐴 ꞷ = − 1 𝜋 𝐼𝑚 𝐺(ꞷ + 𝑖𝛿) 𝑘 𝛥 ꞷ + 𝑖𝛿 = ꞷ − 𝐺 −1 • Next, integrate with quantum impurity solvers 𝛴 = 0
- 23. Hands-on! 23 https://colab.research.google.com/github/knc6/jarvis-tools-notebooks/blob/master/jarvis-tools-notebooks/Qiskit_based_electronic_bandstructure_.ipynb https://github.com/knc6/jarvis-tools-notebooks/blob/master/jarvis-tools-notebooks/Qiskit_based_electronic_bandstructure_.ipynb Can be used for: • Qiskit • Tequila • Pennylane,…
- 24. 24 Summary & future work • A robust workflow for applying quantum algorithms to simulate solids • Applied to ~1000 solids (expanding QC beyond H2, LiH etc.) • Open-source package, contributions are welcome • Opportunities for DMFT/Defect calculations • Challenges: 1) finding minimal circuit, 2) impurity quantum solvers https://jarvis.nist.gov Email: kamal.choudhary@nist.gov https://github.com/usnistgov/jarvis https://github.com/usnistgov/atomqc @dr_k_choudhary