# Towards the simulation of transition-metal oxides of the cathode battery
materials using VQE methods

Marwa H. Farag<EMAIL_ADDRESS>Joydip Ghosh  [

###### Abstract

Variational quantum eigensolver (VQE) is a hybrid quantum-classical technique
that leverages noisy intermediate scale quantum (NISQ) hardware to obtain the
minimum eigenvalue of a model Hamiltonian. VQE has so far been used to
simulate condensed matter systems as well as quantum chemistry of small
molecules. In this work, we employ VQE methods to obtain the ground-state
energy of LiCoO2, a candidate transition metal oxide used for battery
cathodes. We simulate Li2Co2O4 and Co2O4 gas-phase models, which represent the
lithiated and delithiated states during the discharge and the charge of the
Li-ion battery, respectively. Computations are performed using a statevector
simulator with a single reference state for three different trial
wavefunctions: unitary coupled-cluster singles and doubles (UCCSD), unitary
coupled-cluster generalized singles and doubles (UCCGSD) and k-unitary pair
coupled-cluster generalized singles and doubles (k-UpCCGSD). The resources in
terms of circuit depth, two-qubit entangling gates and wavefunction parameters
are analyzed. We find that the k-UpCCGSD with k=5 produces results similar to
UCCSD but at a lower cost. Finally, the performance of VQE methods is
benchmarked against the classical wavefunction-based methods, such as coupled-
cluster singles and doubles (CCSD) and complete active space configuration
interaction (CASCI). Our results show that VQE methods quantitatively agree
with the results obtained from CCSD. However, the comparison against the CASCI
results clearly suggests that advanced trial wavefunctions are likely
necessary to capture the multi-reference characteristics as well as the
correlations emerging from high-level electronic excitations.

###### keywords:

Battery materials, Active space, VQE, UCCSD, UCCGSD, k-UpCCGSD

] Research and Advanced Engineering, Ford Motor Co., Dearborn, MI 48124, USA

##  1 Introduction

Li-ion batteries (LIBs) are chemical energy storage devices, and they are the
dominant propulsive energy source for electric vehicles (EVs)1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14. Given the accelerated rate of EVs market
penetration globally, it is critical that the energy source that will power
these EVs is well understood in order to meet global demand. While LIBs have
been the front runner in leading this battery revolution, it is clear that
further progress in battery chemistry is needed to not only improve the
battery range but also do so under all performance, comfort and user
experience requirements as they are afforded by gas powered vehicles.
Additionally, it is also clear that alternate battery chemistry variants are
critical in order to balance the expected demand and the availability of
critical battery materials.

To further advance the battery technology, we need to improve the battery
materials in terms of their energy density, power density, life-cycle, safety,
cost, and quite importantly recyclability15, 16, 17, 18, 19. Computational
chemistry can provide insights about the charge/discharge mechanisms,
electrochemical and thermal stability, structural phase transition, and
surface behavior20, 21, 22, 23, 24, 25, 26, and it plays a vital role to find
potential materials that can enhance the battery performance and robustness.

To solve the many-body electronic structure problem and find the energy
eigenstate and the corresponding energy eigenvalue, it is essential to choose
a method that allows the best trade-off between accuracy and efficiency27.
This is, however, very challenging for simulations on classical processors
because exact method such as full configuration interaction (FCI) 28 scales
factorially $\mathcal{O}(N!)$, while coupled-cluster methods such as CCSD and
CCSD(T)29 scale as $\mathcal{O}(N^{6})$ and $\mathcal{O}(N^{7})$,
respectively, where $N$ is the number of orbitals30, 31. The unfavorable
scaling of these methods makes their application intractable for large
systems, such as solid state materials.

Current state-of-the-art methods employ density functional theory (DFT) to
simulate solid-state materials32, 33, 34. This method scales as
$\mathcal{O}(N^{3})$ on classical computers, which makes it practical on
today’s supercomputers. However, DFT has some fundamental limitations35, 36,
37, 38, 39, 40. For example, it is known that DFT fails, even qualitatively,
for systems with strong electron correlation41, 42, 43, 44. Another limitation
of the DFT is the self-interaction energy error40. That makes its application
to certain classes of systems challenging, such as systems with transition
metals45, 46. Transition metals are one of the main component in the cathode
battery materials. Previous theoretical works47, 48 employed different DFT
functionals to predict the band structure of the cathode battery materials. It
has been shown that DFT functionals, such as PBE and SCAN 47, 48, cannot
reproduce the experimental values of the band gap for some use cases. For
example, the reported experimental band gap for the LiMnO2 is 1.6449 eV, while
the PBE and SCAN functionals47 give a band gap of 0.92 and 1.19 eV,
respectively. Analyzing the band gap plays an important role to assess the
electronic conductivity of the materials. PBE and SCAN functionals were also
employed to estimate the voltage50, 51, 47. Results have shown that for PBE
functional, the voltage profile of LiMnO2 ranges from 2.4-3.2 V and for SCAN
functional, the profile ranges from 2.9-3.4 V47. On the other hand, the
experimental voltage profile ranges from 2.0-4.6 V52. Estimating the voltage
of the electrodes accurately is essential for understanding the battery
stability. It is known that for LIB, the voltage of the electrode must be
within the voltage window of the electrolyte. Although treating the self-
interaction energy error by using the DFT+U approach53, 54 shows better
agreement with the experimental value of both the band gap and voltage47, the
adjustable U parameter is a species-specific, i.e., the U parameter needs to
be recalculated for each material and for every oxidation state for a given
material55. Hybrid functionals can, also, be used to reduce the self-
interaction energy error by evaluating the exchange energy partly using
Hartree Fock method which comes at a computational overhead56, 57. However, as
shown in previous theoretical works, hybrid DFT functionals do not always work
well to compute the properties of transition metal systems46, 58, 59.
Furthermore, the long range characteristics of the electrostatic potential
makes it computationally demanding to evaluate the exact exchange energy in
periodic calculations60 and, therefore, only short range contribution is
usually considered61, 57, 50. Another issue with the DFT method is the non-
covalent interactions37, 62 which can be corrected using functionals such as
DFT+V63 and OPTPBE-VDW64.

Accurate and universal tools are needed to simulate battery materials and
guide the development of the next generation LIBs, in particular when the
experimental results are unavailable. Quantum computing is an emerging new
computational paradigm for a wide class of problems that are intractable to
existing high performance computers65. Quantum chemistry simulation, such as
the electronic structure problem of molecules and materials, is one of the
most promising near-term applications of quantum computers66, 67, 68, 69. It
is expected that quantum computing will allow accurate simulation of chemical
systems for industry-relevant length scales. There are two categories of
algorithms typically used to perform electronic structure calculations on
quantum computers in order to find the ground-state wavefunction and the
corresponding energy eigenvalue of a given Hamiltonian. The first algorithm is
the quantum phase estimation (QPE)70, which requires quantum error correction
and, therefore, is only amenable to a fault-tolerant quantum computer (FTQC).
The second one is the variational quantum eigensolver (VQE)71 that can be
implemented on the near-term quantum devices, such as noisy intermediate scale
quantum (NISQ) hardware.

A recent theoretical work72 developed a quantum algorithm based on QPE to find
the ground-state energy for cathode battery materials and estimate the cell
voltage, ionic mobility, and thermal stability. It was reported that under the
assumption of single reference state (Hartree Fock approximation), thousands
of logical qubits and trillions of logical gates are required to execute QPE.
Another theoretical work73 estimated the resources required to simulate
electrolyte molecules of LIB on a photonic FTQC. The authors of this work
suggest architectural and algorithmic techniques to allow use of multiple
“magic state fraction” to produce magic state in parallel. The authors found
that for PF${}^{-}_{6}$ using cc-pVDZ basis a total of 16382 logical qubits
and $2.32\times 10^{12}$ T gates are needed. Clearly, the application of these
methods will not be possible on the near-term quantum devices.

In this paper, we simulate cathode battery materials using VQE methods and
benchmark its performance against the classical wavefunction-based techniques.
We simulate gas-phase models of the cathode battery material LiCoO2. To
estimate the performance of the VQE-method, we compare the results obtained
from the VQE simulations with those obtained from the complete active space
configuration interaction (CASCI), complete active space self-consistent field
(CASSCF), coupled-cluster with singles and doubles excitation (CCSD), and the
second-order Møller-Plesset perturbation theory (MP2) on classical
computers30, 31. CCSD scales as $\mathcal{O}(N^{6})$ ($N$ is the number of
orbitals), while CASCI and CASSCF scales exponentially with the size of the
active space orbitals. MP2 scales as $\mathcal{O}(N^{5})$. It is important to
note that a legitimate (classical) benchmark of VQE must include some
wavefunction approach so that we can compare the chemical accuracies for a
given number of orbitals and electrons.

The rest of the paper is organized as follows. First, we discuss the
difference between VQE-UCC trial wavefunctions that are employed in this work
and the active space Hamiltonian. Then, we show how the gas-phase model was
defined to mimic the building block of the crystal structure of the material
and how the active space is selected for the subsequent VQE simulations.
Afterward, we explain the basic steps of the VQE simulations as well as the
details of computational methods. Next, we present the calculated energies and
discuss the difference between the calculated energies using different trial
wavefunctions with the aim of understanding the performance and the resources
needed for simulating battery materials on the near-term quantum devices.
Finally, we summarize our finding.

##  2 Theory

VQE is an iterative quantum-classical variational algorithm. Here, the quantum
device is used to prepare a parameterized wavefunction
$|\Psi(\bm{\theta})\rangle$, while the classical device is used to optimize
the parameters $(\bm{\theta})$ over a relevant cost function. In the
electronic structure problem, the quantum device prepare the wavefunction and
measures the expectation value of the Hamiltonian ($\hat{H}$), while the
classical device is used to optimize the vector parameters $(\bm{\theta})$ and
calculate the energy,
$\langle\Psi(\bm{\theta})|\hat{H}|\Psi(\bm{\theta})\rangle$, where
$\hat{H}=\sum_{i}\hat{h}_{i}$74.

The preparation of the wavefunction consists of two steps. The first step is
the construction of the reference state $|\phi_{0}\rangle$ on the classical
device, while the second step is the preparation of the trial wavefunction on
the quantum computer. For the second step, three different trial wavefunctions
based on the unitary coupled-cluster (UCC) ansatz are employed (UCCSD, UCCGSD,
and k-UpCCGSD)75. The unitary version of coupled cluster (CC) method is suited
on quantum computers due to the properties of the applied gate operation.
Furthermore, it is worth mentioning that in contrast to the conventional CC on
classical computers which is not variational, the UCC on quantum devices is
variational.

In the unitary coupled cluster with singles and doubles excitation (UCCSD),
the trial wavefunction is written as76

$|\Psi(\bm{\theta})\rangle=e^{\hat{T}{(\bm{\theta})}-\hat{T}^{\dagger}(\bm{\theta})}|\phi_{0}\rangle$
(1)

where the singles and doubles cluster operators
$(\hat{T_{1}}(\bm{\theta})-\hat{T_{1}}^{\dagger}(\bm{\theta}))$,
$(\hat{T_{2}}(\bm{\theta})-\hat{T_{2}}^{\dagger}(\bm{\theta}))$, respectively,
are given by

$\displaystyle\hat{T}_{1}(\bm{\theta})-\hat{T}_{1}^{\dagger}(\bm{\theta})$
$\displaystyle=\sum_{i,m}\theta_{i}^{m}\left(\hat{a}_{m}^{\dagger}\hat{a}_{i}-h.c.\right)$
(2) $\displaystyle\hat{T}_{2}(\bm{\theta})-\hat{T}_{2}^{\dagger}(\bm{\theta})$
$\displaystyle=\frac{1}{4}\sum_{i,j,m,n}\theta_{i,j}^{m,n}\left(\hat{a}_{m}^{\dagger}\hat{a}_{n}^{\dagger}\hat{a}_{i}\hat{a}_{j}-h.c.\right)$
(3)

where $i,j$ denote the occupied orbitals and $m,n$ label the unoccupied
orbitals in the reference state $|\phi_{0}\rangle$. $h.c.$ is the Hermitian
conjugate. Here, the excitation is allowed from occupied orbitals to
unoccupied orbitals.

In the generalized singles and doubles UCC (UCCGSD), the trial wavefunction
ansatz is given by77

$|\Psi(\bm{\theta})\rangle=e^{\hat{T}{(\bm{\theta})}-\hat{T}^{\dagger}(\bm{\theta})}|\phi_{0}\rangle$
(4)

and the generalized singles and doubles excitation operators are defined as

$\displaystyle\hat{T}_{1}(\bm{\theta})-\hat{T}_{1}^{\dagger}(\bm{\theta})$
$\displaystyle=\sum_{p,q}\theta_{p}^{q}\left(\hat{a}_{q}^{\dagger}\hat{a}_{p}-h.c.\right)$
(5) $\displaystyle\hat{T}_{2}(\bm{\theta})-\hat{T}_{2}^{\dagger}(\bm{\theta})$
$\displaystyle=\frac{1}{4}\sum_{p,q,r,s}\theta_{p,q}^{r,s}\left(\hat{a}_{r}^{\dagger}\hat{a}_{s}^{\dagger}\hat{a}_{p}\hat{a}_{q}-h.c.\right)$
(6)

Unlike the UCCSD, the generalized singles and doubles excitation in the UCCGSD
does not distinguish between occupied and unoccupied orbitals in the reference
state. Therefore, $p,q,r,s$ indices represent both types of the orbitals,
occupied and unoccupied.

The $k$ unitary pair with generalized singles and doubles product wavefunction
is written as77

$|\Psi(\bm{\theta})\rangle=\prod_{\delta=1}^{k}\left(e^{\hat{T}^{\delta}{(\bm{\theta})}-\hat{T}^{\dagger\delta}(\bm{\theta})}\right)|\phi_{0}\rangle$
(7)

where $k$ is an integer number that increases the wavefunction flexibility. In
k-UpCCGSD, the two-body excitation is allowed to move a pair of electrons from
one spatial orbital to another spatial orbital,

$\hat{T}_{2}(\bm{\theta})-\hat{T}_{2}^{\dagger}(\bm{\theta})=\frac{1}{4}\sum_{p,q}\theta_{p_{\alpha},p_{\beta}}^{q_{\alpha},q_{\beta}}\left(\hat{a}_{q_{\alpha}}^{\dagger}\hat{a}_{q_{\alpha}}^{\dagger}\hat{a}_{p_{\beta}}\hat{a}_{p}-h.c.\right)$
(8)

In Table 1, we introduce the cost of the three UCC trial wavefunctions on
quantum devices using a single Trotter step. As seen, k-UpCCGSD requires the
least and the UCCGSD requires the most resources. In the result and discussion
section, we explain the performance of these methods to simulate gas phase
models, Li2Co2O4/Co2O4, which mimic the building block of the cathode battery
material.

Table 1:  The cost of the UCCSD, UCCGSD, and k-UpCCGSD using a single Trotter
step on quantum devices.

Method | gate count _a_ | circuit depth _b_
---|---|---
UCCSD |  $\mathcal{O}$ $(M-N)^{2}$ $M^{2}$ |  $\mathcal{O}$ $(M-N)^{2}M$
UCCGSD |  $\mathcal{O}$ ($M^{4}$) |  $\mathcal{O}$ ($M^{3}$)
k-UpCCGSD |  $\mathcal{O}$ ($kM^{2}$) |  $\mathcal{O}$ ($kM$)

_a_ Gate count refers to the total number of quantum gates.
_b_ Circuit depth is the number of sequential steps to execute one- and two-
qubit gates.
_c_ $M$ denotes the number of spin-orbitals and $N$ represents the number of
electrons.

Once the trial wavefunction is constructed, the total energy of the system can
be obtained as74

$E=\mathrm{min}_{\bm{\theta}}\,\frac{\langle\Psi(\bm{\theta})|\hat{H}|\Psi(\bm{\theta})\rangle}{\langle\Psi(\bm{\theta})|\Psi(\bm{\theta})\rangle}$
(9)

using the variational principle. Here, $\hat{H}$ is the electronic
Hamiltonian. In the second quantization formalism, the Hamiltonian in the
Hartree Fock (HF) basis can be written as30, 31

$\hat{H}=\sum_{pq=1}^{N}h_{pq}\hat{a}_{p}^{\dagger}\hat{a}_{q}+\frac{1}{2}\sum_{pqrs=1}^{N}h_{pqrs}\hat{a}_{p}^{\dagger}\hat{a}_{q}^{\dagger}\hat{a}_{r}\hat{a}_{s}$
(10)

where $h_{pq}$ and $h_{pqrs}$ correspond to the one and two electron
integrals, respectively

$\displaystyle h_{pq}$ $\displaystyle=\int
d\sigma\,\psi_{p}^{*}(\sigma)\left(-\frac{\nabla^{2}_{\bm{r}}}{2}-\sum_{i}\frac{Z_{i}}{|\bm{R}_{i}-\bm{r}_{i}|}\right)\,\psi_{q}(\sigma)$
(11) $\displaystyle h_{pqrs}$ $\displaystyle=\int\int
d\sigma_{1}d\sigma_{2}\frac{\psi_{p}^{*}(\sigma_{1})\psi_{q}^{*}(\sigma_{2})\psi_{r}(\sigma_{1})\psi_{s}(\sigma_{2})}{|\bm{r}_{1}-\bm{r}_{2}|}$
(12)

where $\bm{R}$ and $\bm{r}$ denote nuclear and electronic spatial coordinate,
respectively. $\sigma$ represents the spatial and spin coordinate and
$\psi(\sigma)$ is a one-electron function that is obtained from HF.

To reduce the circuit depth and overcome the limited qubits coherence time on
NISQ devices, the active space approach is employed28, 78, 31. In this
approach, the Fock space of the wavefunction is divided into active (A) and
inactive (I) orbitals

$|\phi_{0}\rangle=|\phi_{0}^{A}\rangle\otimes|\phi_{0}^{I}\rangle$ (13)

The active orbitals include the highest occupied molecular orbitals and the
lowest unoccupied molecular orbitals in a system. The assumption behind this
approach is that the wavefunction is relatively dominanted by small number of
determinants that one can capture by expanding the wavefunction in the active
space. As a consequence, one can reduce the Hilbert space of the UCC ansatz to
search to a subspace

$|\Psi(\bm{\theta})\rangle=U(\bm{\theta})\,|\phi_{0}^{A}\rangle$ (14)

where $U(\bm{\theta})$ corresponds to one of the ansatzes described above.
Here, the cluster operator allows only excitations among active orbitals. In
this case, the active space Hamiltonian is given by79

$\hat{H}=E_{I}+\sum_{tu}(t|\hat{f}|u)\,\hat{t}^{\dagger}\hat{u}+\frac{1}{2}\sum_{tuvw}(tu|vw)\,\hat{t}^{\dagger}\hat{v}^{\dagger}\hat{w}\hat{u}$
(15)

where $t,u,v,w$ denote the active orbitals. The scalar term $E_{I}$ is the HF
energy of the inactive orbitals. The one-electron integral is given by

$(t|\hat{f}|u)=(t|\hat{h}|u)+\sum_{i}2(tu|ii)-(ti|ui)$ (16)

where $i$ represents an inactive orbital. The spatial one and two-electron
integrals are given in chemists’ notation.

##  3 The gas-phase model and the active space

Figure 1:  (a) The conventional unit cell of the LiCoO2. (b) The $2\times
2\times 1$ supercell of the LiCoO2. (c) The gas-phase model of the Li2Co2O4
(d) The gas-phase model of Co2O4 after removing two-lithium ions and two
electrons from Li2Co2O4. (e) The natural orbital occupation numbers (NOON) as
obtained from MP2/cc-pvdz for the Li2Co2O4 and (f) Co2O4 gas-phase models.

In this work, we employ gas-phase models that mimic the building blocks of the
LiCoO2/CoO2 crystal structure. The gas-phase models are generated based on the
crystal structure of LiCoO2 (R$\bar{3}$m) which we obtained from the Material
Project data base55,https://materialsproject.org. In Fig.1(a), we present the
conventional unit cell that includes three formula units of LiCoO2. Fig.1(b)
shows the LiCoO2 supercell comprises $2\times 2\times 1$ conventional unit
cells and 12 formula units. We notice that every formula unit consists of one
Cobalt (Co), two Oxygens (O), and one Lithium (Li). In the lithiated crystal,
Co is $+3$, Li is $+1$ and O is $-2$. Therefore, the overall charge of LiCoO2
is zero. In the delithiated crystal (CoO2), the Li ion is removed and one
electron is released. In this case, Co becomes $+4$ and the overall charge
stays zero.

To identify a small gas-phase model system for the simulation on quantum
devices that mimics the LiCoO2/CoO2 crystal structure building block, the
following criteria should be met: First, the overall charge should be
maintained (for LiCoO2/CoO2 it is zero); second, the local oxidation state
should be preserved (Co ($+3,+4$), O($-2$), and Li ($+1$)); and third, the
local coordination environment for the transition metal (Co) should be
conserved (every Co is coordinating with two oxygen). Evidently, these
criteria represent the formula unit of LiCoO2/CoO2. In this work, we identify
a gas-phase model which involves two formula units of the LiCoO2/CoO2 and they
are depicted in Fig.1(c) and Fig.1(d). We should emphasize that these gas
phase models do not represent the LiCoO2/CoO2 bulk material; however, they are
useful starting point for estimating the performance and cost of VQE-UCC
methods on transition metal oxides for battery materials.

The constructed gas-phase model of the Li2Co2O4 has 92 electrons and 170
molecular orbitals, while the Co2O4 has 86 electrons and 142 molecular
orbitals using the cc-pVDZ basis set31. Obviously, it is impossible to
simulate these systems using the FCI approach on classical processors.
Furthermore, we cannot explicitly simulate all electrons of these systems on
quantum computers due to the limited number of qubits available in the
existing quantum hardware. Hence, we decompose the Fock space into active and
inactive orbitals and the active space is employed as an initial guess for the
subsequent simulation on a quantum computer. This approach allows reduced
circuit depth making it amenable to the currently available quantum computers.

Figure 2:  The natural orbitals and the corresponding natural occupation
number of the gas-phase model Li2Co2O4 as obtained from MP2/cc-pVDZ.
Jmolhttp://jmol.sourceforge.net is used to plot the orbitals with cutoff=0.05.
Figure 3:  The natural orbitals and the corresponding natural occupation
number of the gas-phase model Co2O4 as obtained from MP2/cc-pVDZ. Jmol is used
to plot the orbitals with cutoff=0.05.

To find the active space $|\phi_{0}^{A}\rangle$, the natural orbitals
occupation numbers (NOON) are analyzed and the results are summarized in Fig.1
(e) for Li2Co2O4 and Fig.1 (f) for Co2O4. The NOON are obtained from the
diagonalization of the one-particle density matrix (1-PDM). The 1-PDM
calculation is carried out using MP2/cc-pVDZ 30, 31, 83 available in the
`Pyscf` package84, 85, 86. MP2 accounts for the correlation of the
wavefunction using perturbation theory, and therefore, it is inexpensive.
Fig.1 (e) and Fig.1 (f) reveal that there are orbitals that are fully occupied
with two-electrons, orbitals that are fully unoccupied (zero electrons), and
orbitals that are semi-occupied (non-integer numbers of electrons). The
orbitals with fractional occupations are considered as the active space, and
the orbitals with integer occupations are considered inactive. As a
consequence, the active space, in this work, comprises of 10 electrons and 10
orbitals (20 spin orbitals). These key orbitals are presented in Fig.2 and
Fig.3. As seen, the active space is dominanted by the Co $d$-orbitals and the
O $p$-orbitals. Previous experimental and theoretical works87, 88, 89, 47
investigating the band structure and the density of state for the crystal
structure of LiCoO2/CoO2 reported that the valence and the lowest conduction
bands are dominant by O $2p$ and Co $3d$ orbitals, and they suggested a strong
hybridization between the O $2p$ and Co $3d$ orbitals. We notice that our
electronic structure analyses for the gas-phase model of Li2Co2O4 in Fig.2 and
Co2O4 in Fig.3 agree well with the electronic structure of the LiCoO2/CoO2
crystals. Therefore, we believe that the gas phase models are a good
approximation for our subsequent VQE simulation. In this work, the natural
orbitals of the active space are employed as an initial guess for the
subsequent simulation on the quantum devices. As known, a linear
transformation between natural orbitals and molecular orbitals exists via the
unitary transformation matrix obtained from the diagonalization of the 1-PDM.

Larger gas-phase models are tested as well to ensure that the results are
consistent with the small gas phase models and the results are presented in
Fig.4. In these models, some of the Co ions are coordinated by six oxygen
atoms. This mimics to some extent the chemical environment of Co in the bulk
materials of LiCoO2. We find that in these larger gas-phase models, the active
space is larger, however, the active space is still dominanted by the O $p$
and Co $d$ orbitals in agreement with the small gas-phase model. Due to the
resource limitations, we only considered the small gas-phase models for the
VQE-UCC simulations in this work.

Figure 4:  The natural orbital occupation numbers (NOON) as obtained from
MP2/6-31g for larger gas-phase models.

##  4 Computational Details

Figure 5:  Schematic representation of the variational quantum eigensolver
algorithm.

Fig.5 outlines the basic steps for our simulation using VQE algorithm. In this
work, all VQE simulations are carried out using `InQuanto` packageA. Tranter,
C. Di Paola, D. M. Ramo, D. Z. Manrique, D. Gowland, G. Greene-Diniz, G.
Christopoulou, I. Polyak, I. Khan, J. Pilipczuk, J. Kirsopp, K. Yamamoto, M.
Tudorovskaya, M. Krompiec, N. Fitzpatrick, InQuanto - Introduction to the
InQuanto Computational Chemistry Platform For Quantum Computers, 2022,
https://medium.com/cambridge-quantum-computing/4fced08d66cc. The charge and
the spin ($2S+1$) for all calculations are set to zero and one, respectively.
The VQE simulation steps can be summarized as follows:

  * •

Classical pre-processing. In this step, we perform all the calculations on
classical processors using `pyscf` package84, 85, 86 along with `InQuanto`
interface. The cc-pVDZ is employed as the basis set for all the classical pre-
processing simulations. We begin by selecting the active space (see the
previous section for more details) by analyzing the natural orbitals
occupation number (NOON), which obtained from the diagonalization of the 1-PDM
using MP2/cc-pVDZ. Next, we calculate and store the necessary matrix elements
for one- and two-electron integrals of the Hamiltonian, see Eq.(15), using RHF
for the subsequent VQE simulations. Finally, we generate the initial guess for
the vector parameter $\bm{\theta}$. Here, all initial parameters are set to
zero.

  * •

Fermion-to-qubit mapping. Jordan-Wigner91, 70 (JW) is employed to perform a
mapping of the second-quantized operators in the Hamiltonian, see Eq.(15), to
qubits. Here, the Hamiltonian is written as a linear combination of string of
Pauli matrices. Similarly, the unitary operator ($U(\bm{\theta})$) of the UCC
wavefunction (see Eq.14) is transformed following the same procedure of the
Hamiltonian. A single Trotter decomposition step is employed for all
calculations.

  * •

Quantum simulations on quantum devices. The trial wavefunction on the quantum
device is constructed in this step. In this work, three different ansatzes
based on the UCC (UCCSD, UCCGSD, k-UpCCGSD) are employed, as explained above
in the theory section. Once the wavefunction is prepared, the expectation
value of the Hamiltonian is computed. All calculations are carried out on the
IBM statevector simulatorhttps://quantum-
computing.ibm.com/lab/docs/iql/manage/simulator/ employing 20 qubits, as our
active space consists of 10 electrons and 20 spin orbitals.

  * •

Parameter optimization. The vector parameters $\bm{\theta}$ are updated using
the bounded limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS-B)
algorithm. An analytical gradient function of the objective function
(expectation value of the Hamiltonian) for UCCSD and k-UpCCGSD is employed,
while for UCCGSD a numerical gradient is employed.

  * •

Convergence. A stopping criterion, based on energy convergence, is selected.
We export the energy once convergence is achieved. Otherwise, the updated
parameters of the wavefunction $\bm{\theta}$ are used to generate a new trial
wavefunction and recalculate the energy. We repeat the process until
converges.

In addition to the VQE-UCC simulations, CASCI, CASSCF, CCSD, and MP2
calculations on classical processors are carried out to benchmark the VQE
results. All calculations are performed using `pyscf` package and cc-pVDZ
basis set. The active space employed in CASCI, CASSCF, CCSD, MP2 is as same as
those selected for the VQE-UCC. In CCSD, only single and double excitations
are considered within the active space. Hence, CCSD span the same Hilbert
space as VQE-UCC trial wavefunctions employed in this work. In CASCI and
CASSCF, FCI is applied within the active space and, therefore, all
configurations within the active space are treated on an equal footing. Unlike
CASCI where only the CI coefficients are optimized variationally, in CASSCF
both orbitals and CI coefficients are optimized variationally to give a lowest
energy for the wavefunction. In MP2, a second-order perturbation theory for
many-electron wavefunction is considered.

##  5 Results and Discussion

In LIBs, the average Li intercalation potential (voltage) is calculated as50

$\langle
V\rangle\simeq\frac{-\left[E(\mathrm{Li}_{x2}\mathrm{MO}_{2})-E(\mathrm{Li}_{x1}\mathrm{MO}_{2})-(x_{2}-x_{1})E(\mathrm{Li})\right]}{(x_{2}-x_{1})e}$
(17)

where $E$ is the total energy and $e$ is the absolute value of the electron
charge. $E(\mathrm{Li}_{x1}\mathrm{MO}_{2})$ and
$E(\mathrm{Li}_{x2}\mathrm{MO}_{2})$ denote the total energy of the material
in different oxidation states during the redox reaction. $E(\mathrm{Li})$ is
the metallic (body-centered-cubic) lithium. In LiCoO2 cathode battery
materials, $E(\mathrm{Li}_{x1}\mathrm{MO}_{2})$ can be LiCoO2, where
$x_{1}=1$, and $E(\mathrm{Li}_{x2}\mathrm{MO}_{2})$ can be CoO2, where
$x_{2}=0$. This will result in voltage per formula unit. When calculating the
energies using DFT, errors such as the self interaction energy error and the
lack of dispersion interaction will not be canceled out, as energies here
represent systems in different environments. Accordingly, DFT predicts the
voltage qualitatively as reported by previous theoretical works50, 25, 20.

To guide the development of novel battery materials with superior performance
in the absence of experimental results, accurate theoretical tools are needed
that enable bridging the gap between accuracy and efficiency. Quantum
computing promises to bridge this gap. In this work, we aim to understand the
performance of the existing VQE-UCC approaches to simulate battery materials
on the near-term quantum devices. This is a step toward establishing a
universal theoretical tools to be employed as a reference value in the absence
of experimental data.

Table 2:  The calculated energy difference (eV) between the lithiated and
delithiated gas-phase models using VQE-UCC ansatz on the IBM statevector
simulator using 20 qubits. The predicted energies are benchmarked against
energies obtained from CASCI, CASSCF, CCSD, and MP2 on classical computers.

Method |  $E$ (eV)_a_ |  $dE$ (kcal/mol) |  $dE$ (kcal/mol)
---|---|---|---
CASSCF | \- 410.8311026 | -0.252199 | 0.054356
CASCI | \- 410.8201662 | 0.0 | 0.306555
CCSD | \- 410.8334597 | -0.306555 | 0.0
MP2 | \- 412.2219177 | -32.325149 | -32.018594
RHF | \- 409.9148477 | 20.87713 | 21.18369
UCCSD | \- 410.8122999 | 0.181401 | 0.487956
UCCGSD | \- 410.9984998 | -4.112470 | -3.805916
k-UpCCGSD (k=3) | -410.7601677 | 1.383598 | 1.690153
k-UpCCGSD (k=5) | -410.8672987 | -1.08690 | -0.780347

_a_ The energy difference between the lithiated and delithiated gas-phase
models
$E(\mathrm{Li}_{2}\mathrm{Co}_{2}\mathrm{O}_{4})-E(\mathrm{Co}_{2}\mathrm{O}_{4})$.

Table 2 shows the energy difference between the Li2Co2O4 and Co2O4 gas phase
models as obtained from UCCSD, UCCGSD, and k-UpCCGSD. The results obtained
from the VQE simulation are compared against those obtained from RHF, CASCI,
CASSCF, CCSD, and MP2 on classical computers. Here, the Li2Co2O4 and Co2O4
gas-phase models mimic the lithiated $E(\mathrm{Li}_{x1}\mathrm{MO}_{2})$ with
$x_{1}=2$ and delithiated $E(\mathrm{Li}_{x2}\mathrm{MO}_{2})$ with $x_{2}=0$
in the voltage calculation, see Eq.(17). In this work, we focus on the energy
difference between Li2Co2O4 and Co2O4 and not the voltage, because the voltage
is usually calculated and measured for bulk materials instead of a simple gas-
phase model. Here, the goal is to quantify the performance and predict the
resources of the VQE-UCC approaches to simulate transition metal systems on
quantum devices. We notice that the energy difference obtained from MP2 and
UCCGSD is the lowest, while the energy difference obtained from RHF is the
highest.

To understand the results in Table 2, we compute the absolute energy value for
our gas phase models and the results are presented in Table 3. We begin by
comparing the VQE-UCC results with the CCSD and CASCI results. In case of
Li2Co2O4, we find that the ground state (GS) energy obtained from RHF is
higher than those obtained from CASCI and CCSD by 186.45 kcal/mol and 182.51
kcal/mol, respectively. This is expected because in RHF, the wavefunction is
described as a one single determinant30. Consequently, the dynamic and static
correlations are missing. On the other hand, the GS energy obtained from the
VQE-UCC ansatz agrees better with CCSD than those obtained from CASCI. As
seen, the GS energy obtained from UCCSD differs from those obtained from CASCI
and CCSD by 3.97 kcal/mol and 0.02 kcal/mol, respectively. Unlike UCCSD,
k-UpCCGSD (k=3) results exihibt less agreement with the results obtained from
CCSD and CASCI. We find that the GS energy obtained from k-UpCCGSD (k=3) is
higher than the GS energy obtained from CASCI and CCSD by 7.17 kcal/mol and
3.22 kcal/mol, respectively. However, adding more k product (k-UpCCGSD (k=5))
brings the results of k-UpCCGSD closer to CCSD and CASCI, because it enhances
the wavefunction flexibility (see Eq. (7)). We find that the GS energy
obtained from k-UpCCGSD (k=5) differs from those obtained using CASCI and CCSD
by 3.46 kcal/mol and -0.48 kcal/mol, respectively. Surprisingly, we notice
that the GS energy obtained from UCCGSD is higher than those obtained from
CASCI and CCSD by 5.56 kcal/mol and 1.62 kcal/mol, respectively. This is
unexpected as UCCGSD wavefunction is more flexible than UCCSD. One reason for
this could be due to the employment of a numerical gradient to calculate the
objective function (expectation value of the Hamiltonian) for the optimization
process during the VQE simulation because of the resource limitations. The
numerical gradient is likely increasing the numerical error. In contrast to
UCCGSD, in UCCSD and k-UpCCGSD, we employ analytical solution to calculate the
objective function. Table 3 also introduces the GS energy values of the Co2O4.
As we can see, similar results to Li2Co2O4 are observed in Co2O4.

Table 3:  The calculated ground state energy in hartree for Li2Co2O4 and Co2O4
using VQE-UCC on IBM statevector simulator and RHF, CCSD, MP2, CASCI, and
CASSCF on classical processors.

Method | E(Li2Co2O4) | dE(kcal/mol) | dE(kcal/mol) | E(Co2O4 ) | dE(kcal/mol) | dE(kcal/mol)
---|---|---|---|---|---|---
CASSCF | -3077.021047220 | -112.146059 | -116.089857 | -3061.923289747 | -111.893859 | -116.144213
CASCI | -3076.842331137 | 0.0 | -3.943799 | -3061.744975570 | 0.0 | -4.250354
CCSD | -3076.836046295 | 3.94379928 | 0.0 | -3061.738202201 | 4.250354 | 0.0
MP2 | -3076.920237209 | -48.886809 | -52.830607 | -3061.771368249 | -16.5616596 | -20.8120138
RHF | -3076.545202892 | 186.450829 | 182.507030 | -3061.481117150 | 165.573694 | 161.323340
UCCSD | -3076.836011187 | 3.965830 | 0.022030 | -3061.738944700 | 3.784429 | -0.465925
UCCGSD | -3076.833469294 | 5.560892 | 1.617092 | -3061.729560089 | 9.673362 | 5.423008
k-UpCCGSD (k=3) | -3076.830912759 | 7.165142 | 3.221343 | -3061.735762095 | 5.781544 | 1.531190
k-UpCCGSD (k=5) | -3076.836817725 | 3.459719 | -0.484080 | -3061.737730070 | 4.546620 | 0.296267

Transition metal oxides, such as the simple gas-phase model employed in this
paper, are known to be strongly correlated systems41, 42, 43, 93, 94. In our
results, we observe that at equilibrium geometry, the results obtained from
CCSD are slightly higher than those obtained from the CASCI by 3.94 kcal/mol
and 4.25 kcal/mol in Li2Co2O4 and Co2O4, respectively. In VQE-UCC, we notice
that the energy difference between UCCSD and CASCI is 3.97 kcal/mol and 3.78
kcal/mol in Li2Co2O4 and Co2O4, respectively. Similar values are observed for
k-UpCCGSD (k=5). To further understand these results, we compare them against
those obtained from MP2 and CASSCF. We note that the CASCI and CCSD GS
energies are higher than CASSCF GS energy by 112.15 kcal/mol and 116.09
kcal/mol, repectively. In addition, the CASCI and CCSD GS energies excess MP2
GS energies by 48.89 kcal/mol and 52.83 kcal/mol, respectively. The
qualitative agreement between VQE-UCC and CASCI, CASSCF, and MP2 could be due
to two reasons. First, the missing correlation from three and more electron
excitations in the cluster operator95, 96, 97; and second, the multireference
character of the wavefunction98, 97, 95. These speculations are valuable
starting points for further studies in the future.

Now, we analyze the energy difference between the lithiated and delithiated
gas-phase models, see Table 2. Here, we expect that error cancellation will
improve the agreement between VQE-UCC method and CASSCF and CASCI. Indeed, in
Table 2, we notice that except for UCCGSD, the values obtained from CCSD and
VQE-UCC are quantitatively agree with those obtained from CASCI and CASSCF.
For instance, the energy obtained from CCSD is slightly lower than that
obtained from CASCI by 0.31 kcal/mol and CASSCF by 0.05 kcal/mol, and in
UCCSD, the energy obtained is slightly above that obtained from the CASCI by
0.18 kcal/mol and CASSCF by 0.43 kcal/mol. Similarly, in k-UpCCGSD (k=5), the
values are slightly lower than that obtained from CASCI by 1.09 kcal/mol and
CASSCF by 0.83 kcal/mol. In Table2, the error cancellation is responsible for
the better performance of the CCSD and VQE-UCC methods with respect to CASCI
and CASSCF.

Table 4:  resource estimation to simulate the gas-phase model Li2Co2O4 and
Co2O4 on the IBM statevector simulator using 20 qubits.

Method | Circuit depth _a_ | Circuit gates | Cnot gates | Parameters
---|---|---|---|---
UCCSD | 32806 | 99910 | 20950 | 675
UCCGSD | 118604 | 344470 | 78150 | 2955
k-UpCCGSD (k=3) | 7113 | 43390 | 5850 | 405
k-UpCCGSD (k=5) | 11765 | 72310 | 9750 | 675

_a_ Number of steps required to execute one-qubit and two-qubit gates on the
circuit.

Next, we investigate the cost and resources needed to simulate Li2Co2O4 and
Co2O4 gas-phase models using VQE-UCC approaches under the assumption of single
reference state on quantum devices. Table 4 presents the resources in terms of
circuit depth, circuit gates, cnot gates and the number of the wavefunction
parameters $\theta$. Circuit depth and gates are important metrics for the
NISQ devices due to the limited coherence time of existing qubits. Our
analyses using 20 qubits show that k-UpCCGSD has the lowest circuit depth and
number of gates compare to the UCCSD and UCCGSD. Results in Table 2 and Table
4 indicate that k-UpCCGSD (k=5) produces results that are similar to UCCSD but
at lower cost. As seen, the circuit depth for the k-UpCCGSD are 11765 compared
to 32806 in UCCSD. Furthermore, the number of gates needed in k-UpCCGSD (k=5)
wavefunction ansatz is 72310 compared to 99910 in UCCSD. For the wavefunction
parameters, we notice that both UCCSD and k-UpCCGSD (k=5) require the same
number of parameters.

Figure 6:  Upper panel: The calculated energy of the Co2O4 using the UCCSD and
k-UpCCGSD (k=5) using VQE method and the conventional CCSD and CASCI on the
classical computer. Lower panel: The difference between the calculated energy
using UCCSD, k-UpCCGSD(k=5) and CCSD.

Finally, we analyze the energy as a function of the distance between the two
formula units. In Fig.6, the potential energy curve of the Co2O4 as a function
of the distance between the two formula units is presented. The equilibrium
distance, here, is 2.8Å. We compare the results obtained from the UCCSD and
k-UpCCGSD (k=5) with those obtained from CASCI and CCSD. Fig.6 uncovers that
the potential energy curve obtained from CCSD and VQE-UCC qualitatively agree
with that obtained from CASCI. This is likely due to the two reasons explained
above. When comparing the VQE-UCC and CCSD results, we find that at short
distance (2.5-4.0 Å) between the two formula units, the results obtained from
VQE-UCC vary almost by 0.5 kcal/mol from that obtained using CCSD. However, at
longer distance (4.5-5.0 Å) the results vary significantly ($>1$ kcal/mol)
with respect to CCSD. One reason for this could be the open-shell character of
the Co+4, when the two Co ions become far from each other. In Co2O4, the Co
ion is ${+4}$ and the electronic configuration is [Ar] $3d^{5}$ ($t_{2}g$: 5
and $e_{g}$: 0). At longer distance, a multi-reference state as an initial
guess is likely needed. This will be a study for future work.

Overall, our results reveal that the VQE-UCC (UCCSD and k-UpCCGSD) approaches
cannot reproduce quantitatively the CASCI and CASSCF GS energy for Li2Co2O4
and Co2O4 gas-phase models and this is likely because UCCSD and k-UpCCGSD do
not capture all the correlation in the total wavefunctions. Employing
multireference wavfunctions as initial guesses and extension of VQE-UCC to
three and more electron excitations are likely to improve the results of VQE-
UCC approaches. In quantum computers we can take advantage of the quantum
nature of the hardware (superposition, entanglement, and interference) to
advance the VQE-UCC and implement the missing correlation and at the same time
keep the VQE-UCC at reasonable scaling. Moreover, our results suggest that we
need 20 qubits to simulate two formula units of LiCoO2 and CoO2. Replicating
these two formula units along the $x$, $y$, and $z$ direction to generate a
supercell of 40 formula units might require approximately 400 qubits that is
likely to be available in the near future. Therefore, VQE method is likely to
be a crucial tool to simulate strongly correlated systems of large size and
provide insights as the quantum hardware matures.

##  6 Conclusions

We simulated Li2Co2O4 and Co2O4 gas-phase models to estimate the performance
of the VQE-UCC (UCCSD, UCCGSD, kUpCCGSD) approaches and compute the resources
needed to simulate cathode battery materials on near-term quantum devices.
Unlike H2, LiH, N2, and H2O that have been employed as benchmark systems to
assess the performance of the VQE method in earlier works, we employed the
transition metal-oxide systems, Li2Co2O4 and Co2O4, which are known to be
strongly correlated systems. In this study, we compared the GS energy obtained
from the VQE-UCC (UCCSD, UCCGSD, kUpCCGSD) approaches with those obtained from
classical wavefunction-based approaches, such as CASCI, CASSCF, CCSD, and MP2.
We found that while the GS energy obtained from VQE-UCC quantitatively agree
with those obtained from CCSD, they only qualitatively agree with those
obtained from CASCI, CASSCF, and MP2. The reasons could be due to the missing
correlation from high-level electronic excitations in the cluster operator and
the multireference character of the wavefunction. Since the single reference
state, such as HF state, has no sufficient overlap with the true ground state
for these systems, an entangled reference state that captures the states
contributing to the total wavefunction might be necessary. Therefore, more
advanced trial wavefunctions need to be developed to simulate such strongly
correlated systems. Future works will focus on developing methods that allow
simulating transition metal systems and capture the missing correlations.

The authors thank Devesh Upadhyay and Andrew Getsoian for fruitful discussion
and for reviewing the manuscript.

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