Quantum simulations of materials on near-term quantum computers

He Ma; Marco Govoni; Giulia Galli

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Quantum simulations of materials on near-term quantum computers

He Ma, Marco Govoni, Giulia Galli

arXiv:2002.11173v1Feb 25, 2020cond-mat.mtrl-sciphysics.chem-phquant-ph

TL;DR

Calculations of materials using DMET and DMFT have been limited to relatively small unit cells. This paper proposes a DFT-based quantum embedding theory for active regions and reports excellent agreement between quantum-computer calculations and classical techniques.

Problem
DMET and DMFT calculations of materials have been limited to relatively small unit cells, of a few tens of atoms.
Source: Embedding Theory · p. 4
Method
The paper proposes a first-principles, DFT-based quantum embedding theory that builds an effective Hamiltonian for an active space.
Source: Introduction · p. 3, Embedding Theory · p. 4, Discussion · p. 8
Results
Quantum-computer calculations showed excellent agreement with classical techniques, with relatively small errors despite quantum-hardware noise.
Source: Discussion · p. 9
Takeaways & Limitations
The work reports first theoretical predictions of singlet states for SiV in diamond and Cr in SiC.
Source: Discussion · p. 9
Exact calculations require approximations in practice to make them feasible.
Source: Introduction · p. 2

Abstract

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Quantum computers hold promise to enable efficient simulations of the properties of molecules and materials; however, at present they only permit ab initio calculations of a few atoms, due to a limited number of qubits. In order to harness the power of near-term quantum computers for simulations of larger systems, it is desirable to develop hybrid quantum-classical methods where the quantum computation is restricted to a small portion of the system. This is of particular relevance for molecules and solids where an active region requires a higher level of theoretical accuracy than its environment. Here we present a quantum embedding theory for the calculation of strongly-correlated electronic states of active regions, with the rest of the system described within density functional theory. We demonstrate the accuracy and effectiveness of the approach by investigating several defect quantum bits in semiconductors that are of great interest for quantum information technologies. We perform calculations on quantum computers and show that they yield results in agreement with those obtained with exact diagonalization on classical architectures, paving the way to simulations of realistic materials on near-term quantum computers.

Introduction

The paper introduces a DFT-based quantum embedding theory for strongly correlated active regions and applies it to spin defects in solids. It also demonstrates quantum-computer simulations that agree with classical calculations.

With NISQ-era qubit limits preventing straightforward simulations of realistic materials, the approach treats strongly correlated active regions at highest accuracy and describes their environment with mean-field theory.Realistic materials may require hundreds of atoms and thousands to millions of degrees of freedom for electronic wavefunctions.Source: Introduction · p. 2
The scalable DFT-based embedding theory includes environment exchange-correlation effects on active regions, going beyond commonly adopted approximations.Source: Introduction · p. 3
For NV [24] [27] [28], SiV [33], and Cr4+ [35, 36, 37] defects, results agree with measurements and predict SiV/Cr singlet positions and ordering, interpreting previously unexplained experiments.Source: Introduction · p. 3
Quantum-computer simulations of NV-center states using quantum phase estimation (PEA) [46] [8] and VQE [48] [49] agree with classical full configuration interaction (FCI) calculations.The calculations use an effective Hamiltonian derived from the embedding theory.Source: Introduction · p. 3

Results

The section applies quantum embedding and quantum simulations to spin defects, assessing agreement with classical FCI results and experimental observations while predicting defect electronic structures and optical cycles.

General strategy

The strategy identifies strongly correlated active regions in a structural model, builds an effective Hamiltonian that includes environmental effects, and solves it classically or with quantum algorithms.Source: General strategy · p. 3

Embedding Theory

The embedding theory treats strongly correlated states in realistic heterogeneous materials, scales to thousands of electrons without evaluating virtual states, and includes exchange-correlation effects beyond RPA.The theory is applied to systems with hundreds of atoms and evaluates exchange-correlation effects with a finite-field algorithm.Source: Embedding Theory · p. 4

Embedding theory applied to spin-defects

The calculations confirm experimental observations for spin defects and describe previously unreported electronic structures that shed light on their optical cycles.The study applies the approach to NV and SiV centers in diamond and Cr in 4H-SiC.Source: Embedding theory applied to spin-defects · p. 5
For NV in diamond, FCI yields the correct symmetry and ordering of the low-lying 3A2, 3E, 1E, and 1A1 states, with exchange-correlation improving agreement with experiment.Source: Embedding theory applied to spin-defects · p. 6
The predicted first optically allowed SiV state is 3Eu at 1.59 eV, near the 1.31 eV ZPL plus 0.258 eV Stokes shift.The paper reports that exchange-correlation effects raise the excitation energies slightly beyond RPA.Source: Embedding theory applied to spin-defects · p. 6
Calculations predicted a 3A2u state 11 meV below 3Eu, supporting Green et al.’s experimental 3A2u-3Eu manifold with a 7 meV energy separation [33]; geometry relaxation may explain the difference.Source: Embedding theory applied to spin-defects · p. 6
For SiV, the predicted 1A1u excitation lies slightly above the 3A2u–3Eu triplet manifold, suggesting energetically unfavorable intersystem crossing to singlets; higher 1Eu and 1A2u states are not expected to matter for the optical cycle.Source: Embedding theory applied to spin-defects · pp. 6–7
For SiV, symmetry forbids first-order ISC from the lowest 1Eg singlet to the 3A2g ground state, suggesting higher-order population and low spin selectivity in the optical cycle.The predictions illuminate strongly correlated SiV singlets and may explain difficulties measuring ODMR, but definitive conclusions require studies including spin-orbit coupling.Source: Embedding theory applied to spin-defects · p. 7
The reported comparison indicates that correlation effects are in better agreement with the measured zero-phonon-line (ZPL) energy of 1.19 eV.The Stokes energy is expected to be small given the large Debye-Waller factor[37].Source: §1.09 · p. 7

1.09 (0.86) eV based on embedding calculations beyond (within) the RPA. Results including exchange-

Cr in 4H-SiC is predicted to have 3E + 3A1 near 1.4 eV and 3E′ + 3A′ near 1.7 eV, resembling experimentally observed Cr-in-GaN manifolds at 1.2 and 1.6 eV.Source: §1.09 · p. 7
Agreement between FCI results and experiment demonstrates that the embedding theory describes strongly correlated regions while retaining a quantitatively correct environment.Source: §1.09 · p. 7

Quantum simulations of spin-defects

PEA converges to classical FCI results as ancilla qubits increase, and hardware VQE reaches ground-state energies within 0.2 eV despite noise.Quantum-simulator VQE also converges for both MS = 1 and MS = 0 ground states.Source: Quantum simulations of spin-defects · p. 8

Discussion

The discussion presents a quantum embedding framework that treats active regions and their environments at different levels of accuracy, enabling simulations of materials on near-term quantum computers. It reports agreement between classical and quantum calculations for semiconductor spin defects and highlights the method’s broader applicability.

Discussion

The embedding treats active regions and their environments at different accuracy levels, using hybrid density functional theory for the environment and effective active-space interactions that include environmental screening and exchange-correlation effects.The whole system is treated quantum mechanically.Source: Discussion · p. 8
Unlike the commonly used random phase approximation, the method includes exchange-correlation effects and applies to heterogeneous materials and large systems using response-function algorithms [68, 69].The discussion identifies the omission of exchange-correlation effects as a limitation of the random phase approximation.Source: Discussion · p. 8
The flexible framework can incorporate dynamical screening, electron-phonon coupling, and a self-consistent environmental screening cycle when the active region influences the host electronic structure.Dynamical screening uses a frequency-dependent screened Coulomb interaction; phonon effects enter through phonon contributions to screened interactions.Source: Discussion · pp. 8–9
Classical and quantum algorithms agreed excellently for semiconductor spin defects, while selected quantum-computer results matched simulator results with relatively small error despite hardware noise.The discussion also reports that treatment of dielectric screening beyond the random phase approximation accurately predicted excitation energies.Source: Discussion · p. 9
The calculations predicted previously unobserved excited states, including the first theoretical singlet-state predictions for SiV in diamond and Cr in SiC, informing their full optical cycle.These are described as the first theoretical predictions of those singlet states.Source: Discussion · p. 9
The method enables NISQ simulations of strongly correlated states in systems with hundreds of atoms, extending prior quantum-embedding studies [17] [18] [19] [20] [21] to defective solids of interest for quantum technologies.The discussion contrasts these solids with pristine materials.Source: Discussion · p. 9
The strategy is applicable to strongly or weakly correlated active regions and can address molecules, materials, catalysts, drugs, and aqueous solutions containing complex dissolved species.Different regions can be treated with varying levels of accuracy.Source: Discussion · p. 9

Methods

The methods combine density-functional calculations with an embedded effective Hamiltonian, then use quantum algorithms to simulate a minimum NV-center model.

Density Functional Theory

Ground-state geometries are relaxed with spin-unrestricted PBE calculations; NV and SiV in diamond use 216-atom supercells, and Cr in 4H-SiC uses 128 atoms.The calculations use a 50 Ry kinetic-energy cutoff and relax until forces are below 0.013 eV/Å.Source: Density Functional Theory · pp. 9–10

Construction of effective Hamiltonians

WEST constructs effective Hamiltonians from spin-restricted DFT wavefunctions, representing response functions and fxc with the first 512 eigenpotentials of χ0.Finite-field fxc calculations couple WEST with Qbox, and PySCF is used for FCI calculations on the effective Hamiltonian.Source: Construction of effective Hamiltonians · p. 10
A dielectric-dependent hybrid functional supplies defect-level mean-field descriptions, while PBE is used for geometries.The hybrid functional self-consistently sets exact exchange from the host dielectric constant; the passages distinguish its role in defect levels from PBE geometry relaxation.Source: Construction of effective Hamiltonians · p. 10, Density Functional Theory · p. 10