††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Quantum Simulations of Material Properties on Quantum Computers

Nan Sheng  Department of Chemistry, University of Chicago, IL 60637, USA.
Christian Vorwerk  Pritzker School of Molecular Engineering, University of
Chicago, Chicago IL 60637, USA  Marco Govoni<EMAIL_ADDRESS>Materials
Science Division and Center for Molecular Engineering, Argonne National
Laboratory, Lemont IL 60439, USA  Pritzker School of Molecular Engineering,
University of Chicago, IL 60637, USA  Giulia Galli<EMAIL_ADDRESS>Pritzker School of Molecular Engineering and Department of Chemistry,
University of Chicago, Chicago IL 60637, USA  Materials Science Division and
Center for Molecular Engineering, Argonne National Laboratory, Lemont IL
60439, USA

###### Abstract

We discuss computational frameworks to carry out electronic structure
calculations of materials on noisy intermediate scale quantum computers using
embedding theories and effective many-body Hamiltonians. The latter are
defined and diagonalized by performing hybrid computations on classical and
quantum architectures. We present applications for a specific class of
materials, i.e., spin-defects in solids, which are promising systems to build
future quantum technologies, e.g., computers, sensors and devices for quantum
communications.

##  I Introduction

Quantum simulations of the physical properties of molecules and solids on
classical computers are routinely used to tackle many problems in materials
science and chemistry [1, 2, 3, 4]; these simulations are aimed at
understanding a variety of complex heterogeneous systems, in diverse fields
such as catalysis [5, 6, 7] and quantum information science [8, 9, 10], as
well as at generating data for computations based on machine learning and
artificial intelligence [11, 12]. The use of quantum computers promises to
improve the efficiency of quantum simulations and to permit the adoption of
higher level theories for systems more complex than tractable at present [13,
14, 15, 16].

A fundamental step in the calculations of properties of molecules and solids
at the quantum mechanical level of theory is the solution of the time
independent Schrödinger equation describing interacting electrons in the field
of the nuclei; such solution provides the basis for the evaluation of numerous
properties, including total and bond energies, excitation energies and
optimized geometries. A common strategy [17, 18, 19, 20] adopted on a quantum
computer is that of writing the electronic structure problem in terms of
second-quantized Hamiltonians, whose parameters are determined on a classical
computer from quantum chemistry [21] or density functional (DFT) and many body
perturbation theory [22, 23] calculations. The Hamiltonian is then represented
in terms of qubits and quantum gates; following simulation schemes for
fermions on a lattice, recent applications used the Jordan–Wigner transform
[24], or the Verstraete-Cirac mapping [25] or the Bravyi–Kitaev transform
[26], which are implemented in several open-source codes (e.g., Qiskit [27],
OpenFermion [28]). Finally, an algorithm is chosen to find the lowest
eigenvalues of the Hamiltonian, compatible with the number of available qubits
and circuit depths of the hardware. Topics of active research include the
study of the scaling of various algorithms, e.g., variational quantum
eigensolver (VQE) [29, 30] and quantum phase estimation [31] with specific
attention to the dimensionality of the problem and whether shallow or deep
circuits are used [32].

Progress has been reported in the past decades in solving the Schrödinger
equation for molecular systems on quantum computers [33, 34, 29, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] ranging from
simple ions, including H3+ and HHe+ to molecules such as water, alkali
hydrides and hydrogen chains with up to 12 atoms, for which Hartree-Fock
solutions have been recently reported [50]. An interesting summary of
calculations of molecular systems on quantum computers is given in Ref. [54],
which also discusses the algorithms used to solve the molecular Hamiltonian in
second quantization.

So far, the majority of studies of condensed systems on quantum computers have
focused on model systems, for example Hubbard and Heisenberg models and one-
dimensional periodic chains [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72], with few investigations of solids using quantum
embedding theories [73] or tight-binding models [74, 75]. Embedding theories
[76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 73, 86, 87, 88, 89, 90, 91, 92, 93,
94, 95, 96, 97] are particularly promising frameworks both from an algorithmic
and application point of view. They are prone to the definition of second-
quantized Hamiltonians for sub-sets of the electronic states in a solid and
hence to the formulation of the problem which is similar to and can draw from
the formulation adopted for molecular systems. Furthermore there are many
interesting problems that naturally lend themselves to a quantum embedding
description, for example spin-defects in solids or in general point defects in
materials, active site of catalysts, molecular adsorbates on surfaces and
nanostructures embedded in condensed systems, including solvents, to name a
few.

In this perspective we discuss frameworks to carry out quantum mechanical
calculations for solids on near-term quantum computers using embedding
theories with focus on a specific class of materials, i.e., spin-defects in
solids (see Fig. 1). These are promising materials to build future quantum
technologies, e.g., computers, sensors and novel devices for quantum
communications. We first present several quantum embedding theories useful to
define effective Hamiltonians and then discuss general strategies to simulate
materials for future quantum technologies using hybrid quantum-classical
calculations, with some specific examples on spin qubits. We conclude with a
discussion of current open challenges.

Figure 1:  From materials to devices and back again: the bottom and top arrows
indicate quantum simulations of materials carried out on a classical or
quantum computer, respectively, leading to the prediction and design of
components for quantum architectures; the latter may then be used to perform
quantum simulations (as indicated by the middle arrow) and in turn improve
predictive capabilities for materials and devices.

##  II Quantum Embedding Theories

Figure 2:  An overview of quantum embedding theories used in condensed matter
physics and quantum chemistry. Density-based theories encompass density-
functional-theory-in-density-functional-theory (DFT-in-DFT) as well as
wavefunction-in-DFT (WF-in-DFT) embedding [76, 77, 78] schemes. Density-matrix
embedding theory (DMET) [79, 80, 81, 82, 83, 84] employs the density matrix to
define an embedding protocol. Green’s function-based methods include quantum
defect embedding theory (QDET) [85, 73, 86], self-energy embedding theory
(SEET) [87, 88, 89], dynamical mean-field (DMFT)+GW [90, 91, 92, 93, 94, 95],
and DMFT+LDA [96, 97] embedding.

Quantum embedding methods are theoretical frameworks to solve the time-
independent Schrödinger equation for a system of electrons by separating the
problem into the calculation of the energy levels or density of a so-called
active space (or “fragment”, “impurity”) and those of the remaining
environment. Each part of the system is described at the quantum mechanical
level [98, 99], with the active space being treated with a more accurate and
computationally more expensive theoretical method than the environment [98].
In contrast, quantum embedding models describe only the active space with
quantum-mechanical methods, while employing a classical description for the
environment [99, 100, 101, 102, 103].

The three key ingredients of a quantum embedding theory are the strategy used
to partition the full system into active space and environment, the
computational method adopted to describe the two portions, and the
approximation for the interaction between active space and environment [104].
As illustrated in Fig. 2, we can classify different embedding theories by
identifying the key quantity used to realize the embedding of the active space
[98, 99]. In density functional theory (DFT) embedding, the active space is
defined by a region in real space and the density of the system is partitioned
into that of the active space and of the environment. DFT calculations for the
environment yield an exchange-correlation embedding potential [76, 77, 78]
entering the Schrödinger equation for the active space, which is solved with a
high-level quantum-chemical method. In density matrix embedding theories [79,
80, 81, 82, 83, 84], the active space is again defined by selecting specific
sites in real space. The electronic structure of the active space is
determined by solving a Schrödinger equation at a high level of theory
including additional bath orbitals, which accounts for the interaction with
the environment. This framework is commonly known as the quantum impurity
problem due to its similarity with the Anderson impurity problem [105]. The
bath orbitals are obtained from a low-level calculation of the full system
with an additional one-particle operator, which is constructed to satisfy the
condition that the density matrix obtained with the low- and high-level
methods be identical.

In Green’s function embedding theories, the active space is defined by a group
of single particle states belonging to the Hilbert space of a mean-field
Hamiltonian of the entire system (e.g., a Kohn-Sham or Hartree-Fock
Hamiltonian). The self-energy of the active space is then expressed as the sum
of self-energies evaluated at a high- and low-level of theory, with an
additional double-counting correction term. Green’s function embedding
theories, such as those based on dynamical mean-field theory (DMFT+LDA [96,
97] and DMFT+GW [90, 91, 92, 93, 94, 95]), or self-energy embedding theory
(SEET) [87, 88, 89] differ by the choice of high- and low-level methods and by
the technique used to separate the terms in the total self-energy.

Recently, we have proposed a quantum embedding theory for the calculation of
defect properties in solids [85, 73, 86], which we denote here as quantum
defect embedding theory (QDET). Although applied so far only to color centers
in wide band gap semiconductors, the QDET is general and may be used to
describe a variety of impurities or defects in a condensed system, e.g.,
adsorbates on surfaces, ions or nanostructures in water, catalytic sites at
interfaces. Hence, the term defect here denotes a small guest region embedded
in a large host system. Similar to most of Green’s function based methods, in
QDET the active space is defined by a set of single particle electronic
states, which includes the states localized in proximity of the defect or
impurity and, in some cases, additional single particle states belonging to
the host material. An effective Hamiltonian in second quantization is then
defined, where the potential within the active space includes the effect of
the environment through an effective many-body screening term. The Hamiltonian
is solved using a full configuration interaction (FCI) approach, thus
including correlation effects between the electronic states of the active
space. The screening term can be calculated within the constrained random-
phase approximation (cRPA) [106, 107, 108] or by including explicitly exchange
correlation effects [109, 110]. In principle the environmental screening
entering the QDET is frequency-dependent, but in practice, current
implementation have so far been restricted to static screening.

###  II.1 Comparison between Green’s function based quantum embedding theories

As shown below, while from a theoretical standpoint QDET is closely related to
DMFT+GW and to SEET, these three frameworks target vastly different
applications and systems of different sizes. Before turning to the discussion
of applications and of their implementations on a quantum computer, it is
interesting to understand the connection between QDET, DMFT+GW and SEET.

In the last two decades, quantum embedding theories that combine dynamical
mean-field theory (DMFT) [111, 112, 113, 97] for the description of the active
space, and many-body perturbation theory (MBPT) for the environment have been
successfully employed to describe correlated states of several solids. Within
this scheme, known as DMFT+GW [90, 114], the active space is defined by a
subset of the electronic bands of a crystal, such as the $3d$ bands in
transition metal compounds [115, 114]. These states are mapped onto an
effective impurity problem, which by construction reproduces the Green’s
function of the active space. The effective impurity and the full active space
are connected through a so called hybridization self-energy $\Delta$, which is
found self-consistently such that the impurity Green’s function is identical
to the local Green’s function of the active space. The influence of the
environment on the active space is described within MBPT, using a diagrammatic
expansion of the self-energy ($\Sigma$) in terms of the screened Coulomb
potential $W$. Vertex corrections are usually neglected in the evaluation of
the self-energy, and hence $\Sigma=\mathrm{i}GW$ [116, 117, 118, 119, 120],
where $G$ is the Green’s function [93, 121, 94, 122, 123, 124]. Within
DMFT+GW, the self-energy of the active space is given by the sum of the DMFT
and GW self-energies plus a double-counting term. The latter is meant to
remove the contribution of diagrams that are contained in both the $GW$ and
DMFT self-energies. Additionally, the interaction within the active space is
screened by that of the environment, and evaluated, e.g., by using the
constrained random-phase approximation (cRPA) [106, 107, 108].

If we consider the super-cell of a solid in real space (i.e., we sample
reciprocal space using only the $\Gamma$ point), the mapping of the active
space to the impurity problem is exact and the hybridization vanishes, i.e.,
$\Delta=0$. This specific case is known as the quantum impurity problem in the
atomic limit [91, 125, 126], where a single, non self-consistent solution of
the impurity problem yields the electronic states of the active space.
Assuming the interactions to be instantaneous, the solution of the impurity
problem can be obtained by exact diagonalization (ED), which is equivalent to
full configuration interaction (FCI). Under these specific conditions, the
DMFT+GW embedding and QDET frameworks are identical. We note that while in
DMFT the impurity problem with frequency-dependent interactions can be solved
within the Langrangian formalism using quantum Monte-Carlo solvers [127, 128,
129, 130] or ED [131], but additional approximations to the interactions
within the active space are required for these solvers.

An embedding approach related to DMFT+GW, called DMFT+LDA@$\Gamma$ was
recently proposed for the study of molecules and nanostructures [132, 133,
134, 135], where the environment is described using DFT in the local density
approximation (LDA). In DMFT+LDA, an exact expression for the double-counting
term is not available, and the interaction within the active space is
described using effective Hubbard parameters, U and J.

Finally we compare QDET with the self-energy embedding theory (SEET) [87, 88,
136, 137]. In both cases the active space is defined by a subspace of the full
electronic Hilbert space. In SEET the active space is used to separate the
independent-particle Green’s function $G^{0}$ into that of the active space
and its environment. Using the Dyson equation $G=G^{0}+G^{0}\Sigma G$, where
$G$ is the full Green’s function, one then defines an impurity problem for the
active space, with an explicit hybridization term ($\Delta$) with the
environment. The interaction within the active space is given by the bare
Coulomb potential. The self-energy of the active space is obtained as the sum
of the impurity self-energy, the self-energy computed at a low-level of theory
(e.g., GW [116, 117, 118, 119]) and a double-counting term. The quantum
impurity problem is solved by FCI [88, 136, 138]. To compare SEET+GW and QDET
we consider again a super-cell calculation. In both embedding theories, the
self-energy is obtained as the sum of a quantum impurity self-energy, GW self-
energy, and a double counting term. If the active space is chosen to be a
subset of the quasi-particle states obtained from the GW calculation for the
full system, the hybridization term vanishes; in this case the SEET+GW
framework reduces to the atomic impurity problem for the active space. The
main difference between SEET+GW and QDET resides in the screening of the
interaction in the active space which is chosen to be the bare Coulomb
potential in SEET+GW, and the screened Coulomb interaction within QDET. This
difference also leads to different expressions for the double-counting
correction to the active-space self-energy [88].

Although, as shown above, for specific choices of the active space and
supercells, QDET, DMFT+GW and SEET+GW are closely related, their use and
implementations have been substantially different. DMFT was originally
proposed to solve interacting lattice models, such as the Hubbard and periodic
Anderson model [112, 129]. Therefore, DMFT+GW has traditionally been applied
to the calculation of the properties of bulk crystals, e.g., transition metal
compounds with correlated $3d$ electrons and lanthanides with $4f$ bands [115,
114], including SrVO3 [139, 92, 122, 124], La2CuO4, and NiO [93]. DMFT+GW is
designed to yield ground-state properties, such as total energies and lattice
parameters, and the properties of charged excitations, which are described by
the spectral function of the one-body Green’s function. QDET is instead aimed
at determining neutral excitations within the active space, i.e., the excited
states of a point defect in a host crystal. In QDET, excited states are
readily available through the FCI solution of the impurity problem in the
atomic limit. Another important difference between the applications of QDET
and DMFT+GW is that the former is tailored to describe active regions in a
large bulk environment, i.e., in solids with hundreds or even thousands of
atoms. We briefly describe below the key ingredients of large scale
implementations of QDET.

###  II.2 Large scale calculations with quantum defect embedding

Figure 3:  Layout of materials simulations using the quantum defect embedding
theory (QDET) of Ref. [73, 85, 86] on a classical and quantum computer. In the
second and fourth panel from the left, representative single particle and many
body states are shown, respectively. The term $W_{E}$ (see text) denotes the
screening that the environment exerts on the active space. The effective
Hamiltonian $H_{eff}$ describes the active space (see text).

The main distinctive features of the implementation of QDET proposed in Ref.
[86] are the (i) calculation of the environmental screening ($W^{E}$) without
evaluating virtual single particle states and without inverting dielectric
matrices and (ii) the straightforward use of hybrid functionals in the DFT
description of the environment. The calculation of $W^{E}$ uses only
projections onto occupied single particle orbitals and employs algorithms
developed in the context of GW calculations [140, 73, 86, 141].

The specific steps of a QDET calculation are (see Fig. 3): after performing a
(hybrid) DFT calculations for the full solid, an active space is defined
(e.g., by choosing four localized orbitals in proximity of the NV- center in
diamond, as in the example of Ref. [85]). The environmental polarizability
$\chi^{E}$ is calculated without explicit summations over empty states (for
example using the WEST code [140, 142, 143, 141]). Using $\chi^{E}$ the
environmental screening $W^{E}$ is computed and used in the definition of the
effective potential $v_{ijkl}^{\text{eff}}$ entering the second quantized
Hamiltonian $H^{\text{eff}}$ that describes the active space in the field of
the environment:
$H^{\text{eff}}=\sum_{ij}^{A}t_{ij}^{\text{eff}}a_{i}^{\dagger}a_{j}+\frac{1}{2}\sum_{ijkl}^{A}v_{ijkl}^{\text{eff}}a_{i}^{\dagger}a_{j}^{\dagger}a_{k}a_{l}$.
Note that $W^{E}$ may be evaluated either within the RPA or taking explicitly
into account the derivative of the exchange-correlation potential ($f_{xc}$).
The latter can be calculated by a finite-field approach, using the WEST [140,
142, 143, 141] and Qbox [144] codes coupled in client server mode [109, 110,
145, 141]. Recently we have proposed several ways to include the effect of
$f_{xc}$ in $W_{E}$, and compared the results obtained for the electronic
structure of several defective systems [86]. Finally we note that care must be
exercised in defining the double-counting term entering the QDET. The
development of double-counting corrections fully consistent with
DMFT+GW@$\Gamma$ techniques is in progress.

Having built the effective Hamiltonian of the active space on a classical
computer, we can diagonalize it on the same computer with FCI (e.g., by using
the PySCF package[146]) or on a quantum architecture, and in the next section
we describe examples of diagonalization on a near-term quantum computer.

Figure 4:  Calculation of the energy of the triplet ground state of the NV-
center in diamond ($M_{S}=0$ state in the ${}^{3}A_{2}$ manifold) using a
variational quantum eigensolver (VQE) on a quantum computer and on a noiseless
quantum simulator [73]. The strongly-correlated
$\frac{1}{\sqrt{2}}(|a\bar{a}e_{x}\bar{e}_{y}\rangle+|a\bar{a}\bar{e}_{x}e_{y}\rangle)$
state is obtained starting from: (A) the $|a\bar{a}e_{x}\bar{e}_{x}\rangle$
state, or (B) the $|a\bar{a}e_{x}\bar{e}_{y}\rangle$ state, where $a$,
$e_{x}$, $e_{y}$ (spin-up) and $\bar{a}$, $\bar{e}_{x}$, $\bar{e}_{y}$ (spin-
down) denote the $a_{1}$ and $e$ single particle orbitals. Only the
occupations of the $e$ orbitals is shown. The parity transformation is used
and the qubit Hamiltonian acts on two qubits. The number of variational
parameters in the UCCSD ansatz optimized with the COBYLA algorithm are six in
(A) and one in (B), respectively. Quantum simulations are performed with the
QASM simulator and the IBM Q Valencia quantum computer using the IBM Qiskit
package [27].

##  III Quantum embedding calculations using a quantum computer

We focus here on the application of the QDET to the study of defects in
materials. Specifically, we are interested in the energy levels of strongly-
correlated states in defect spin qubits, given their key role in
initialization and read-out of spin qubits [10, 147, 148, 9, 149, 150, 151]. A
spin defect may be considered as an impurity in a host solid described by a
periodic supercell. In this case, impurity states are not only localized in
space but they also correspond to energy levels usually localized in energy
within the band gap of the solid. The many-body spectrum obtained by
diagonalizing the effective Hamiltonian allows for the description of singlet
and triplet states and of all the necessary transitions involved, e.g., in
inter-system crossings. We have recently applied the QDET to spin qubits in
diamond and silicon carbide [73, 85, 86], with results in good agreement with
experiments [152, 153].

We report in Fig. 4 the results for the ground state of the NV- center in
diamond, as obtained on a quantum computer and a quantum simulator. The full
system is described at the DFT level of theory with a periodic 216-atom
supercell, and using the plane-wave pseudopotential method [73]. The use of
QDET permits the definition of an effective Hamiltonian that acts on the
$a_{1}$ and the $e$ orbitals and whose triplet ground state ${}^{3}A_{2}$ may
be obtained either with exact diagonalization on a classical hardware or with
the VQE algorithm on a quantum computer. Despite the presence of noise in the
quantum hardware, whose full understanding and characterization will be
critical to improve the use of quantum algorithms [154], the results obtained
on a quantum computer (blue lines) converge to the ground state energy given
by exact diagonalization (dotted line). We note that the converged results are
affected differently by the hardware noise, depending on the chosen starting
point in the optimization procedure. In particular we obtain results within a
$\sim 10$ meV or $\sim 130$ meV of the exact diagonalization one, if the
initial state has one or six variational parameters, respectively. By using a
quantum simulator the effect of the noise is not present and, as expected, we
find perfect agreement between the results obtained with VQE (orange line) and
with exact diagonalization (dotted line) on the classical hardware.

Another interesting class of defective systems is that of point defects in
oxides [155, 156]. Studies of these systems have been conducted, for example,
using DMFT+LDA[157] and LDA+U[158] on classical computers. QDET is an
interesting alternative for the investigation of defects in oxides, especially
when a simulation with large supercells is required. As mentioned earlier, in
addition to point defects in solids, ions or nanostructures in water [159,
160] are other types of impurities that are amenable to be studied with the
QDET.

Other embedding theories, in particular DMFT and DMET, have been implemented
on quantum architectures for Hubbard models describing small molecules [66,
67, 68, 69, 70, 51, 71, 72, 52, 53], and VQE or PEA, have been adopted as
many-body solver for the embedded Hamiltonian.

##  IV Outlook

Quantum embedding theories represent useful frameworks to perform quantum
simulations of heterogeneous systems on near term quantum computers. Proof of
principle calculations of the electronic structure of spin qubits in diamond
have been recently presented in the literature, showing the same results on
classical computers and on quantum simulators; these results are consistent
with those obtained on a noisy intermediate scale quantum (NISQ) computers,
which are however affected by the hardware noise. Although convergence is
obtained in all cases (classical and quantum computers and quantum
simulators), the NIQS results for the ground state are slightly higher in
energy, due to the presence of noise. Spin qubits are just one of the possible
applications of quantum embedding theories, which may in principle be applied
to a variety of problems, including solvated ions and nanostructures,
adsorbates on surfaces and catalytic sites at surfaces and interfaces.

However, the verdict is not yet out, on whether quantum computers will
substantially improve the scaling of algorithms in use in classical computers
to diagonalize the Hamiltonian of active sites; in other words it has not yet
been established whether quantum computers will in fact lead to much more
efficient calculations of the time independent Schrödinger equation of
interacting electrons. The evaluation of hard bounds on the performance of the
variational algorithms employed in the presence of noise and combined with
(partial) error-mitigation techniques remain difficult. In addition the type
of pre-factors to be expected to the possible polynomial scaling of quantum
computational algorithms has not yet been established and of course pre-
factors will play an important role in determining for which size of systems
quantum advantage may be eventually reached. Fault-tolerant error correction
schemes, necessary to apply quantum simulations to realistic materials, are an
active and yet relatively new field of research and questions regarding which
kind of schemes might be needed to achieve chemical accuracy are still very
much open. In addition, for all hybrid simulations where data are moved from
classical to quantum computers, communications schemes need to be carefully
engineered in order to avoid transfer bottlenecks. Finally, we note that here
we just briefly addressed the problem of solving the time independent
Schrödinger equation for the electrons in solids, namely a basic electronic
structure problem at fixed ionic coordinates. Computation of excited states,
in addition to ground state is work in progress and calculations of additional
properties of materials, including structural stability, electron-phonon
interaction and finite temperature properties remain as of yet largely
unexplored on quantum computers.

###### Acknowledgements.

We thank Benchen Huang and He Ma for useful discussions. This work was
supported by MICCoM, as part of the Computational Materials Sciences Program
funded by the U.S. Department of Energy. This research used resources of the
National Energy Research Scientific Computing Center (NERSC), a DOE Office of
Science User Facility supported by the Office of Science of the US Department
of Energy under Contract No. DE-AC02-05CH11231, resources of the Argonne
Leadership Computing Facility, which is a DOE Office of Science User Facility
supported under Contract DE-AC02-06CH11357, and resources of the University of
Chicago Research Computing Center.

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