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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).
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
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.”
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
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/
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.
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
![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)