Computing band gaps of periodic materials via sample-based quantum diagonalization

TL;DR

A sample-based quantum diagonalization workflow for simulating electronic states of periodic materials, and for predicting their band gaps, which outperforms select quantum-chemical benchmarks as well as approaches based on density functional theory, the standard reference in materials simulation of solids.

Abstract

A key objective of computational solid state physics is to predict electronic properties of periodic materials. However, electronic structure simulations based on density functional theory fail to predict experimental results if correlations are not properly accounted for. Here, we report a sample-based quantum diagonalization workflow for simulating electronic states of periodic materials, and for predicting their band gaps. To that end, we devise a general lattice Hamiltonian representation in which material-specific, electronic interaction parameters are obtained self-consistently. Two exemplar, wide-gap materials - hafnium dioxide and zirconium dioxide - are expressed as quantum circuits that leverage the lattice representation with a materials-specific parametrization. We sample the quantum circuits on a state-of-the-art, superconducting quantum processor and diagonalize the lattice Hamiltonian in the reduced configuration subspaces with standard techniques. Our method outperforms select quantum-chemical benchmarks as well as approaches based on density functional theory, the standard reference in materials simulation of solids. Importantly, the quantum-computed band gap predictions for the two dielectrics agree with independent lab experiments. In essence, quantum-classical hybrid simulation workflows on pre-fault tolerant quantum computers produce useful, experimentally verifiable property predictions in applied materials science.

Figures (6)

• Figure 1: Hybrid, quantum-classical computational workflow for predicting band gaps of periodic materials. Based on a lattice Hamiltonian representation with materials-specific parametrization, sample-based quantum algorithms compute band gap energies for comparison with experimental results.
• Figure 2: Lattice Hamiltonian representation of periodic materials. A tight-binding Hamiltonian with hopping parameters $t_{ij}$ is complemented with intra-site $U_{i}$ and inter-site $V_{kl}$ interaction parameters, see \ref{['eq:hubbard-hamiltonian']}, that are calculated self-consistently for a specific material within the DFT+U+V framework. The electronic density is projected onto a localized, atomic orbital basis set. Each atomic orbital of every atom in the crystal is mapped onto a single site in the projected lattice, as illustrated here for a representative solid with a cubic lattice.
• Figure 3: SQD and Ext-SQD results obtained on the ibm_pittsburgh quantum computer with circuits containing (a) 38 qubits for HfO2 and (b) 46 qubits for ZrO2, respectively. The qubits representing alpha electrons are highlighted in red, the qubits representing beta electrons in green, and the qubits representing ancilla qubits in blue. (Lower panels) Error as function of configuration space fraction relative to classical diagonalization, for the electron-number subspaces $\{N_e-1, N_e, N_e+1\}$ which are needed to compute the band gap. The respective HCI energies are used as reference.
• Figure 4: Band gap energies at the $\Gamma$-point for (left) HfO2 and (right) ZrO2 obtained with DFT simulations, as well as with the lattice Hamiltonian representation, assuming various levels of electronic interactions: non-interacting electrons (TB), intra-site interactions ($U$), inter-site interactions ($V$), and both intra-site and inter-site interactions simultaneously ($U+V$). The Ext-SQD results were obtained with the ibm_pittsburgh quantum computer. Computational results obtained in GW approximation are plotted as references. The gray shades cover the ranges of experimental band gap values as measured in the lab.
• Figure S1: Results obtained from the projection of the DFT electronic structure. (a) Band structures for Si (top), ZrO2 (middle) and HfO2 (bottom). The Quantum Espresso results using standard DFT are shown in dashed black line, while the PAOFLOW projection is represented by solid red lined. (b) Graphical representation of the Hamiltonian at the $\Gamma$-point for the same materials. The color map represents the normalized squared value of the hopping matrix elements ($|t_{ij}|^2$) and the columns and rows are the localized orbital atomic basis. To allow for a good visualization of the interaction and hopping terms, we do not show the diagonal elements, which are associated with the energy of orbitals.