# Construction of Antisymmetric Variational Quantum States
with Real-Space Representation

Takahiro Horiba<EMAIL_ADDRESS>Soichi Shirai  Hirotoshi Hirai
Toyota Central R&D Labs., Inc., 41-1 Yokomichi, Nagakute, Aichi 480-1192,
Japan

###### Abstract

Electronic state calculations using quantum computers are mostly based on
second quantization, which is suitable for qubit representation. Another way
to describe electronic states on a quantum computer is first quantization,
which is expected to achieve smaller scaling with respect to the number of
basis functions than second quantization. Among basis functions, a real-space
basis is an attractive option for quantum dynamics simulations in the fault-
tolerant quantum computation (FTQC) era. A major difficulty in first
quantization with a real-space basis is state preparation for many-body
electronic systems. This difficulty stems from of the antisymmetry of
electrons, and it is not straightforward to construct antisymmetric quantum
states on a quantum circuit.

In the present paper, we provide a design principle for constructing a
variational quantum circuit to prepare an antisymmetric quantum state. The
proposed circuit generates the superposition of exponentially many Slater
determinants, that is, a multi-configuration state, which provides a
systematic approach to approximating the exact ground state. We implemented
the variational quantum eigensolver (VQE) to obtain the ground state of a one-
dimensional hydrogen molecular system. As a result, the proposed circuit well
reproduced the exact antisymmetric ground state and its energy, whereas the
conventional variational circuit yielded neither an antisymmetric nor a
symmetric state. Furthermore, we analyzed the many-body wave functions based
on quantum information theory, which illustrated the relation between the
electron correlation and the quantum entanglement.

##  I Introduction

Quantum computers are currently attracting increasing attention as promising
hardware for materials computations [1, 2, 3], and a number of studies on a
variety of material systems have been conducted [4, 5, 6, 7, 8, 9, 10, 11, 12,
13]. Such materials computations are mostly based on second quantization [14,
15, 16], which is suitable for describing electronic states on quantum
computers.

An alternative way to describe electronic states on quantum computers is first
quantization, in which a wave function is specified by the expansion
coefficients of the basis functions. With quantum computers, it is possible to
obtain a wave function represented by an exponential number of basis functions
with a polynomial number of qubits [17]. Therefore, first quantization offers
the possibility of achieving smaller scaling with respect to the number of
basis functions than second quantization [18, 19, 20, 21, 22].

Among basis functions, a real-space basis is an attractive option because a
systematic improvement of computational accuracy and a rapid convergence to
the continuum limit can be expected by increasing the number of qubits [23].
In addition, a real-space basis can be applied to systems with a variety of
boundary conditions [24, 23], and thus is suitable for quantum dynamics
calculations [25]. Recently, Chan et al. proposed quantum circuits for
computing real-space quantum dynamics based on first quantization [21], which
represented a promising blueprint of quantum simulations in the fault-tolerant
quantum computing (FTQC) era.

However, first quantization has a significant challenge, namely state
preparation. Let us consider preparing the ground state of the system, which
is a typical choice of initial state for dynamics calculations. As a state
preparation method, we consider quantum phase estimation (QPE) [26] which has
been employed in several studies on first quantization [19, 21]. By using QPE,
it is possible to distill the ground state from an input state which has
sufficient overlap with the ground state [19]. Thus, the problem is how to
prepare such an input state.

In first quantization with a real-space basis, preparing an input state takes
a tremendous number of gate operations because probability the amplitudes of
the many-body wave function need to be encoded into a state vector of a
quantum circuit. Although several amplitude-encoding methods have been
proposed [27, 28, 29, 30], it is not at all straightforward to prepare an
approximate ground state with any of them. This state preparation problem is
often avoided by using oracle circuits. Compared to constructing such oracle
circuits, variational methods such as the variational quantum eigensolver
(VQE) [14, 15, 16] are considered to be relatively feasible approaches. In
fact, a number of studies have proposed preparing an input state using the VQE
in second quantization [31, 32].

Unfortunately, it is also not straightforward to implement the VQE in first
quantization. This is due to the antisymmetry of electrons, which are Fermi
particles. In second quantization, antisymmetry does not need to be considered
for variational quantum states, because antisymmetry is naturally introduced
as the anticommutation relations of creation and annihilation operators. By
contrast, in first quantization, antisymmetry is imposed on the many-body wave
function itself, and thus variational quantum states must satisfy
antisymmetry. Nevertheless, there is no guarantee that quantum states
generated by conventional variational quantum circuits will satisfy
antisymmetry, which raises the possibility that the VQE will yield non-
antisymmetric states. Therefore, a variational quantum circuit that generates
only antisymmetric quantum states is required in order to obtain the
electronic ground state by the VQE.

In the present paper, we provide a design principle for constructing such an
antisymmetrized variational quantum circuit. Our proposed circuit consists of
two types of variational circuits that achieve antisymmetry-preserving
transformations of a state vector on a Hilbert space. It is noteworthy that
the proposed circuit generates a multi-configuration (MC) state. That is, it
is possible to generate superpositions of an exponentially large number of
Slater determinants by alternately layering these two types of variational
circuits. This scheme provides a systematic approach to approximating the
exact ground state. To verify the validity of our method, we performed the VQE
calculation for a one-dimensional hydrogen molecular (1D-$\rm{H_{2}}$) system,
and demonstrated that the proposed circuit well reproduced the exact
antisymmetric, or fermionic, ground state and its energy. In addition to
implementing the VQE, we analyzed the many-body wave functions based on
quantum information theory. Such an analysis reveals the microscopic
electronic structure of a many-body wave function represented in real space
and illustrates the relation between the electron correlation and the quantum
entanglement.

##  II Method

Figure 1: Quantum circuit to generate an antisymmetrized variational quantum
state for one-dimensional two-electron systems

In this section, we introduce the first-quantized VQE framework based on the
real-space representation and describe our proposed variational quantum
circuit. We also describe the setting of the numerical experiments for a
1D-$\rm{H_{2}}$ system.

###  II.1 First-quantized VQE with a real-space basis

To begin with, we briefly describe the first-quantized formulation of the
many-body electron problem. The many-body Schrödinger equation for an
$\eta$-electron molecular system is expressed by the following equation:

$H\ket{\psi(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}=E\ket{\psi(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})},\\\
$ (1)

where $\ket{\psi(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}$ is the
antisymmetric many-body wave function to be constructed on a quantum circuit.
The molecular Hamiltonian $H$ with the Born-Oppenheimer approximation is
expressed in atomic units as follows:

$H=\sum_{i=1}^{\eta}\left[-\frac{{\bm{\nabla_{i}}}^{2}}{2}-\sum_{p}\frac{Z_{p}}{|\bm{r_{i}}-\bm{R_{p}}|}\right]+\sum_{i<j}\frac{1}{|\bm{r_{i}}-\bm{r_{j}}|},\\\
$ (2)

where $r_{i}$ is the $i$th electron coordinate, $R_{p}$ is the $p$th nuclear
coordinate, and $Z_{p}$ is the atomic number of the $p$th nucleus. The VQE in
this study is based on this first-quantized Hamiltonian and the many-body wave
function represented in real space.

Next, we introduce the real-space basis. Let us consider the expansion of the
many-body wave function by the real-space basis
$\ket{\delta(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}$ (delta functions
located at grid points $\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}}$). The
expansion coefficient of a many-body wave function is given as

$\psi(\bm{r^{\prime}_{1}},\bm{r^{\prime}_{2}},\cdots,\bm{r^{\prime}_{\eta}})=\braket{\delta(\bm{r^{\prime}_{1}},\bm{r^{\prime}_{2}},\cdots,\bm{r^{\prime}_{\eta}})}{\psi(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}.\\\
$ (3)

The real-space basis of an $\eta$-electron system consists of that of one-
electron system $\ket{\delta(\bm{r_{i}})}$, as follows:

$\ket{\delta(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}=\ket{\delta(\bm{r_{1}})}\ket{\delta(\bm{r_{2}})}\cdots\ket{\delta(\bm{r_{\eta}})}.\\\
$ (4)

$\ket{\delta(\bm{r_{i}})}$ consists of three-dimensional spatial coordinates
$x,y,z$ and one spin coordinate $s$. Assuming that $L$ qubits are assigned to
each spatial dimension and one qubit to spin, $\ket{\delta(\bm{r_{i}})}$ is
expressed as

$\ket{\delta(\bm{r_{i}})}=\ket{{{x_{1}}}^{(i)}\cdots{{x_{L}}}^{(i)}}\ket{{{y_{1}}}^{(i)}\cdots{{y_{L}}}^{(i)}}\ket{{{z_{1}}}^{(i)}\cdots{{z_{L}}}^{(i)}}\ket{s^{(i)}},\\\
$ (5)

where
${x_{k}}^{(i)},{y_{k}}^{(i)},{z_{k}}^{(i)},s^{(i)}\in\left\\{0,1\right\\},\forall
k\in[1,L]$. The number of qubits that constitute
$\ket{\delta(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}$ is $\eta(3L+1)$,
and thus the many-body wave function is expanded in $2^{\eta(3L+1)}$ basis
functions. Quantum computers are expected to realize such an exponentially
large number of basis functions with a polynomial number of qubits, which is a
significant advantage over classical computers.

In order to implement the VQE, it is necessary to measure the energy
expectation value of a quantum state
$\ket{\psi(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}$. The energy
expectation value of molecular Hamiltonian $H$ in Eq. 2 is expressed as
follows:

$\begin{split}E&=\braket{\psi(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}{H}{\psi(\bm{r_{1}},\bm{r_{2}},\cdots,\bm{r_{\eta}})}\\\
&=E_{K}+E_{V}^{e-n}+E_{V}^{e-e},\end{split}$ (6)

where $E_{K}$ is the electron kinetic energy, $E_{V}^{e-n}$ is the electron-
nuclei Coulomb attraction energy, and $E_{V}^{e-e}$ is the electron-electron
Coulomb repulsion energy. The kinetic and Coulomb energy operators of the
Hamiltonian are diagonal in momentum space and real space, respectively. The
momentum space basis is obtained by the quantum Fourier transformation (QFT)
of the real-space basis. Letting
$\ket{\psi(\bm{k_{i}})}=U_{\mathrm{QFT}}\ket{\psi(\bm{r_{i}})}$ be the one-
body momentum space basis, the kinetic energy $E_{K}$ is expressed as

$E_{K}=\sum_{i=1}^{\eta}-\frac{{\bm{k_{i}}}^{2}}{2}|\psi(\bm{k_{i}})|^{2}.$
(7)

The Coulomb energies $E_{V}^{e-n},E_{V}^{e-e}$ are expressed in terms of one-
body and two-body real-space bases as follows:

$\begin{split}E_{V}^{e-n}&=-\sum_{i=1}^{\eta}\sum_{p}\frac{Z_{p}}{|\bm{r_{i}}-\bm{R_{p}}|}|\psi(\bm{r_{i}})|^{2},\\\
E_{V}^{e-e}&=\sum_{i<j}\frac{1}{|\bm{r_{i}}-\bm{r_{j}}|}|\psi(\bm{r_{i}},\bm{r_{j}})|^{2}.\end{split}$
(8)

Probability distributions $|\psi(\bm{k_{i}})|^{2}$, $|\psi(\bm{r_{i}})|^{2}$,
and $|\psi(\bm{r_{i},r_{j}})|^{2}$ can be obtained by measuring the output of
the (QFT applied) quantum circuit in the computational basis (Pauli $Z$
basis). Note that, as pointed out by Chen et al., this method is not efficient
in terms of sampling cost and is sensitive to discretization errors of grids
[21]. According to their work, QPE should be used for measuring energy
expectation values in future quantum calculations, but this naive method was
adopted in the present study because of a lack of sufficient computational
resources to simulate QPE on a classical computer. The remaining issue in the
first-quantized VQE is how to construct a variational quantum circuit that
generates antisymmetric ansatz states. In the following, we describe the
design principle for constructing a variational quantum circuit to settle the
above issue.

###  II.2 Antisymmetrized variational quantum circuit

To explain the design principles of circuits, we consider the one-dimensional
two-electron system, which is the minimum model required to describe the
antisymmetry of electrons. In the following part of this section , we discuss
two-electron systems, but we would like to note that the proposed method is
applicable not only to two-electron systems but more generally to systems with
larger numbers of electrons.

The overall circuit architecture is shown in Fig. 1. The proposed circuit
consists of three parts: (1) seed state preparation, (2) one-body and two-body
space variational circuits, and (3) measurement of energy expectations. Since
the measurement procedure of energy expectations has already been described,
we here describe only the parts of the circuit that generate antisymmetric
ansatz states. In the first part of the circuit, some antisymmetric state,
i.e., seed state, is prepared, at which the state vector of the circuit
belongs to the antisymmetric subspace. The subsequent one-body and two-body
space variational circuits transform the state vector into the ground state
while keeping it in the antisymmetric subspace. Such a preparation of an
antisymmetric state and antisymmetry-preserving transformations are the main
design principle of the proposed circuit. In the following, we describe each
part of the circuit.

####  II.2.1 Seed state preparation

The simplest antisymmetric state of a two-electron system is the Greenberger-
Horne-Zeilinger (GHZ) state, as follows:

$\ket{\psi_{\mathrm{GHZ}}}=\frac{1}{\sqrt{2}}(\ket{0}_{1}\ket{1}_{2}-\ket{1}_{1}\ket{0}_{2}).$
(9)

This state can be generated by a series of an X (NOT) gate, an H (Hadamard)
gate, and a CNOT (controlled NOT) gate as shown in Fig. 1. In this circuit,
the spin coordinates $\ket{s^{(i)}}$ are antisymmetrized, meaning that this
state is a singlet state in which the two spins are antiparallel
$\downarrow_{1}\uparrow_{2}-\uparrow_{1}\downarrow_{2}$ (0 is assumed to be
$\downarrow$ and 1 to be $\uparrow$). This state is expressed as follows,
including the spatial coordinates (the normalization constant $1/\sqrt{2}$ is
omitted):

$\ket{\psi_{\mathrm{seed}}(\bm{r_{1}},\bm{r_{2}})}=\ket{\bm{0}}_{1}\ket{0}_{1}\otimes\ket{\bm{0}}_{2}\ket{1}_{2}-\ket{\bm{0}}_{1}\ket{1}_{1}\otimes\ket{\bm{0}}_{2}\ket{0}_{2},$
(10)

where the spatial coordinate of each electron is $\ket{\bm{0}}=\ket{00\cdots
0}$. Obviously, this state is antisymmetric under the exchange of two
electrons and can be used as a seed state.

For more-than-two-electron systems, seed states cannot be constructed in such
a simple way. As an example, for three-electron systems, the following Slater
determinant consisting of the three states $\ket{00}$, $\ket{01}$, and
$\ket{10}$ is one of the antisymmetric states (the normalization constant
$1/\sqrt{3!}$ is omitted).

$\begin{split}\ket{\psi_{00,01,10}}&=\begin{vmatrix}\ket{00}_{1}&\ket{01}_{1}&\ket{10}_{1}\\\
\ket{00}_{2}&\ket{01}_{2}&\ket{10}_{2}\\\
\ket{00}_{3}&\ket{01}_{3}&\ket{10}_{3}\\\ \end{vmatrix}\\\
&=\ket{00}_{1}\ket{01}_{2}\ket{10}_{3}+\ket{01}_{1}\ket{10}_{2}\ket{00}_{3}\\\
&\quad+\ket{10}_{1}\ket{00}_{2}\ket{01}_{3}-\ket{10}_{1}\ket{01}_{2}\ket{00}_{3}\\\
&\quad-\ket{01}_{1}\ket{00}_{2}\ket{10}_{3}-\ket{00}_{1}\ket{10}_{2}\ket{01}_{3}.\end{split}$
(11)

Previous research has proposed methods for constructing quantum circuits to
prepare such a Slater determinant, though it is not as simple as that for the
GHZ state. Thanks to the work of Berry et al., an implementation of the
Fisher-Yates shuffle [33] on a quantum circuit has been provided and can
generate the superpositions of an exponential number of permutation elements
with polynomial computational complexity [19]. Following their method, it is
possible to systematically generate seed states for more-than-two-electron
systems.

Another way to prepare a seed state is a variational approach, which achieves
a seed state using conventional variational circuits. In this approach, the
Hadamard test (swap test) can be employed as an objective function. Letting
$\mathrm{SWAP}_{ij}$ be the swap operator acting on the subspace of the $i$th
and $j$th electrons, the output of the Hadamard test for the quantum state
$\ket{\psi}$ becomes

$p_{0}=\frac{1+\braket{\psi}{\mathrm{SWAP}_{ij}}{\psi}}{2},p_{1}=\frac{1-\braket{\psi}{\mathrm{SWAP}_{ij}}{\psi}}{2},$
(12)

where $p_{0}$ and $p_{1}$ are the measurement probabilities of the $\ket{0}$
and $\ket{1}$ of an ancilla qubit. Suppose $\ket{\psi}$ is an antisymmetric
wave function, $\mathrm{SWAP}_{ij}\ket{\psi}=-\ket{\psi}$; then the output of
the Hadamard test becomes $p_{0}=0,p_{1}=1$ (for symmetric wave functions,
$p_{0}=1,p_{1}=0$). Therefore, an approximated antisymmetric state can be
obtained by updating variational parameters to maximize the output of the
Hadamard tests for all two-electron swap operations. By repeatedly performing
the Hadamard test on an approximated antisymmetric state, the pure
antisymmetric states can eventually be distilled, which can then be used as a
seed state.

####  II.2.2 One-body space variational circuit

The proposed variational quantum circuit consists of two components: a one-
body space variational circuit and a two-body ($\eta$-body) space variational
circuit. We first explain the one-body space variational circuit. A one-body
space variational circuit consists of unitary operators that act on the
subspace of each electron (one-body space). Consider
$\mathcal{U}^{(1)}(\theta)=\left[U^{(1)}(\theta)\right]^{\otimes\eta}$ as the
one-body space variational circuit, where $U^{(1)}(\theta)$ is a unitary
operator (variational circuit) acting on a one-body space. Since
$U^{(1)}(\theta)$ transforms a state vector equally in each one-body space,
$\mathcal{U}^{(1)}(\theta)$ does not destroy the antisymmetry of the ansatz
state. For two-electron systems,
$\mathcal{U}^{(1)}(\theta)=U^{(1)}(\theta)\otimes U^{(1)}(\theta)$ acting on
the seed state (GHZ state) $\ket{\psi_{\mathrm{seed}}(\bm{r_{1}},\bm{r_{2}})}$
yields

$\begin{split}\ket{\psi(\bm{r_{1}},\bm{r_{2}})}&=\mathcal{U}^{(1)}(\theta)\ket{\psi_{\mathrm{seed}}(\bm{r_{1}},\bm{r_{2}})}\\\
&=U^{(1)}(\theta)\ket{\bm{0}}_{1}\ket{0}_{1}\otimes
U^{(1)}(\theta)\ket{\bm{0}}_{2}\ket{1}_{2}\\\
&\quad-U^{(1)}(\theta)\ket{\bm{0}}_{1}\ket{1}_{1}\otimes
U^{(1)}(\theta)\ket{\bm{0}}_{2}\ket{0}_{2}\\\
&=\ket{\alpha(\theta)}_{1}\otimes\ket{\beta(\theta)}_{2}-\ket{\beta(\theta)}_{1}\otimes\ket{\alpha(\theta)}_{2}.\end{split}$
(13)

As can be seen, $\mathcal{U}^{(1)}(\theta)$ transforms
$\ket{\bm{0}}\ket{0},\ket{\bm{0}}\ket{1}$ into
$\ket{\alpha(\theta)}=U^{(1)}(\theta)\ket{\bm{0}}\ket{0}$,
$\ket{\beta(\theta)}=U^{(1)}(\theta)\ket{\bm{0}}\ket{1}$ in each one-body
space, and antisymmetry is preserved under this transformation. The resulting
state can be expressed by a single Slater determinant consisting of
$\ket{\alpha(\theta)}$ and $\ket{\beta(\theta)}$ as follows:

$\begin{split}\ket{\psi_{\mathrm{SD}}(\bm{r_{1}},\bm{r_{2}})}&=\ket{\alpha(\theta)}_{1}\otimes\ket{\beta(\theta)}_{2}-\ket{\beta(\theta)}_{1}\otimes\ket{\alpha(\theta)}_{2}\\\
&=\begin{vmatrix}\ket{\alpha(\theta)}_{1}&\ket{\beta(\theta)}_{1}\\\
\ket{\alpha(\theta)}_{2}&\ket{\beta(\theta)}_{2}\end{vmatrix}.\end{split}$
(14)

This indicates that a one-body space variational circuit can only explore
quantum states within the HF approximation. By using the two-body
($\eta$-body) space variational circuit described in the following, we can
explore quantum states beyond the HF approximation.

####  II.2.3 Two-body space variational circuit

The usual strategy to go beyond the HF approximation is based on configuration
interaction (CI) theory, in which a many-body wave function is approximated by
a linear combination, or superposition, of multiple Slater determinants. As
previously mentioned, a one-body space circuit generates an electronic state
expressed by a single Slater determinant. Therefore, a superposition of
multiple one-body space circuits is expected to generate a superposition of
multiple Slater determinants, that is, an MC state.

For a two-electron system, consider two different operators $U_{a}\otimes
U_{a},U_{b}\otimes U_{b}$, where $U_{a}$ and $U_{b}$ are unitary operators
acting on a one-body space . Their superposition is expressed as

$U^{(2)}(\theta)=c_{a}(\theta)\cdot U_{a}\otimes U_{a}+c_{b}(\theta)\cdot
U_{b}\otimes U_{b},$ (15)

where $c_{a}(\theta)$, $c_{b}(\theta)$ are superposition coefficients
parametrized by $\theta$. Operator $U^{(2)}(\theta)$ acting on a single Slater
determinant $\ket{\psi_{\mathrm{SD}}}$ yields

$\begin{split}&U^{(2)}(\theta)\ket{\psi_{\mathrm{SD}}}\\\
&=c_{a}(\theta)\begin{vmatrix}U_{a}\ket{\alpha}_{1}&U_{a}\ket{\beta}_{1}\\\
U_{a}\ket{\alpha}_{2}&U_{a}\ket{\beta}_{2}\\\
\end{vmatrix}+c_{b}(\theta)\begin{vmatrix}U_{b}\ket{\alpha}_{1}&U_{b}\ket{\beta}_{1}\\\
U_{b}\ket{\alpha}_{2}&U_{b}\ket{\beta}_{2}\\\ \end{vmatrix}\\\
&=c_{a}(\theta)\ket{\psi^{a}_{\mathrm{SD}}}+c_{b}(\theta)\ket{\psi^{b}_{\mathrm{SD}}}.\end{split}$
(16)

As expected, a superposition of two different Slater determinants
$\ket{\psi^{a}_{\mathrm{SD}}}$, $\ket{\psi^{b}_{\mathrm{SD}}}$, is generated
by $U^{(2)}(\theta)$.

Figure 2: Implementation of $U^{(2)}(\theta)$. Top: implementations of Ising
gate and real-valued symmetry-preserving (RSP) ansatz. Bottom: conceptual
picture of generation of superposed HF states by an $R_{zz}(\theta)$ gate.

One of the simplest implementations of such an operator is the Ising gate
[34]. Ising gates such as $R_{xx}(\theta)$, $R_{yy}(\theta)$, and
$R_{zz}(\theta)$ are represented by a superposition of $I\otimes I$ and
$P\otimes P$ ($I$ is the identity operator, $P$ is the Pauli operator $X$,
$Y$, $Z$); thus, they can be employed as $U^{(2)}(\theta)$. For example,
$R_{zz}(\theta)$ shown in Fig. 2 is represented by

$\begin{split}R_{zz}(\theta)&=\begin{pmatrix}e^{-i\theta/2}&0&0&0\\\
0&e^{i\theta/2}&0&0\\\ 0&0&e^{i\theta/2}&0\\\ 0&0&0&e^{-i\theta/2}\\\
\end{pmatrix}\\\ &=\cos\frac{\theta}{2}\cdot I\otimes
I-i\sin\frac{\theta}{2}\cdot Z\otimes Z.\end{split}$ (17)

As can be seen from Fig. 2, the Ising gates act on a two-body space; thus, we
refer to $U^{(2)}(\theta)$ as a two-body space variational circuit. For
$\eta$-electron systems, an $\eta$-body space variational circuit
$U^{(\eta)}(\theta)$ is represented by
$U^{(\eta)}(\theta)=\sum_{i}c_{i}(\theta){U_{i}}^{\otimes\eta}$. Such
operators can be easily implemented using Ising gates with cascaded CNOT gates
across the $\eta$-body space [35, 36].

The Ising gate is not the only option as a two-body space variational circuit.
For example, the real-valued symmetry-preserving (RSP) ansatz [37] shown in
Fig. 2 can also be used as a two-body space variational circuit. The RSP
ansatz is represented by the superposition of the following operators:

$\begin{split}U_{\mathrm{RSP}}(\theta)&=\begin{pmatrix}\cos\theta&0&0&-\sin\theta\\\
0&0&1&0\\\ 0&1&0&0\\\ \sin\theta&0&0&\cos\theta\\\ \end{pmatrix}\\\
&=\frac{1}{2}\left[X\otimes X+Y\otimes Y+\cos\theta(I\otimes I+Z\otimes
Z)\right.\\\ &\qquad\left.-i\sin\theta(X\otimes Y+Y\otimes
X)\right].\end{split}$ (18)

Here, terms $X\otimes Y,Y\otimes X$ appear as tensor products of different
operators $X$ and $Y$. Either of these terms alone destroys the antisymmetry
of ansatz states, but pairs of them preserve antisymmetry as follows:

$\begin{split}&(X\otimes Y+Y\otimes X)\ket{\psi_{\mathrm{SD}}}\\\
&=\begin{vmatrix}X\ket{\alpha}_{1}&Y\ket{\beta}_{1}\\\
X\ket{\alpha}_{2}&Y\ket{\beta}_{2}\\\
\end{vmatrix}+\begin{vmatrix}Y\ket{\alpha}_{1}&X\ket{\beta}_{1}\\\
Y\ket{\alpha}_{2}&X\ket{\beta}_{2}\\\ \end{vmatrix}\\\
&=\ket{\psi^{\prime}_{\mathrm{SD}}}+\ket{\psi^{\prime\prime}_{\mathrm{SD}}}.\end{split}$
(19)

In Eq. 18, the RSP ansatz is represented by a real-valued unitary operator,
which is convenient for describing real-valued eigenstates of one-dimensional
systems. Unfortunately, however, it is not obvious how to extend this ansatz
to a more-than-two-body space. Although the Ising gates are considered a more
suitable option for generalizing to a system with a larger number of
electrons, in the present study, the RSP ansatz was employed for convenience.

As illustrated in Fig. 2, a single Slater determinant is split each time
$U^{(2)}(\theta)$ acts on it. Therefore, it is expected that an exponentially
large number of Slater determinants will be generated by repeatedly applying
the two-body space variational circuit followed by the one-body space
variational circuit. However, such consecutive application of two-body
variational circuits does not increase the number of superposed Slater
determinants as one might expect. This is because a product of multiple Pauli
operators reduces to a single Pauli operator due to the commutation relation
among Pauli operators ($P_{i}^{2}=I,P_{1}P_{2}=iP_{3}$). This reduction of
Pauli operators can be prevented by using the alternating layered structure of
one-body and two-body space variational circuits as shown in Fig. 1. If the
one-body space variational circuit is noncommutative with the Pauli operators,
then the reduction of Pauli operators between two-body space variational
circuits is prevented. Therefore, by repeatedly applying one-body and two-body
space variational circuits alternately to a seed state, the number of
superposed Slater determinants can be increased exponentially, and so is a
systematic approach to approximating the exact ground state.

During the optimization process of the VQE framework, variational parameters
in one-body and two-body circuits are simultaneously optimized so as to
minimize the energy expectation values of an ansatz state. As can be seen from
Eqs. 14–16, optimizing one-body and two-body circuits corresponds to
optimizing the electronic states composing each Slater determinant (electronic
orbitals) and the superposition coefficients of the Slater determinants (CI
coefficients), respectively. Therefore, this framework can be regarded as the
multi-configuration self-consistent field (MCSCF) method, which is one of the
post-Hartree-Fock ab initio quantum chemistry methods. A notable advantage of
the proposed method is its ability to perform MCSCF calculations based on an
exponential number of electron configurations with polynomial computational
complexity.

###  II.3 Settings for Numerical Simulations

To verify the validity of our method, we performed the VQE calculation with
the proposed circuit to obtain the ground state of a 1D-$\rm{H_{2}}$ system.
The Hamiltonian of this system is expressed as follows:

$\begin{split}H&=\sum_{i=1}^{2}\left[-\frac{1}{2}\frac{d^{2}}{dr_{i}^{2}}-\sum_{p=1}^{2}\frac{1}{\left|r_{i}-R_{p}+\epsilon\right|}\right]\\\
&\qquad+\frac{1}{|r_{1}-r_{2}+\epsilon|}+\frac{1}{|R_{1}-R_{2}|}.\end{split}$
(20)

where $r_{1},r_{2}$ and $R_{1},R_{2}$ are the positions of electrons and
protons, respectively. To avoid zero division in the Coulomb interaction
terms, a small real value $\epsilon$ is added to the denominators of those
terms (soft Coulomb potential).

Figure 3: Probability amplitudes $\ket{\psi(r_{1},r_{2})}$ of degenerated
ground states (one fermionic state ($S^{2}=0,S_{z}=0$) and three bosonic
states ($S^{2}=1,S_{z}=0,\pm 1$)) at interatomic distance $|R_{1}-R_{2}|=$ 0.5
bohrs ($R_{1},R_{2}$ are shown by pink dots). Each many-body wave function is
shown in the four subspaces corresponding to the spin configurations
$\uparrow_{1}\uparrow_{2},\uparrow_{1}\downarrow_{2},\downarrow_{1}\uparrow_{2},\downarrow_{1}\downarrow_{2}$.
Note that the correspondence between colors and values is different for each
plot.

Note here that the ground states of fermions and bosons are completely
degenerate. Figure 3 shows four degenerate ground states obtained by the exact
diagonalization of Eq. 20 at the interatomic distance $|R_{1}-R_{2}|=$ 0.5
bohrs. As shown in Fig. 3, the ground states of the fermions are the singlet
state ($S^{2}=0,S_{z}=0$) whose spin symmetry is antisymmetric and spatial
symmetry is symmetric. On the other hand, the ground states of the bosons are
the triplet state ($S^{2}=1,S_{z}=0,\pm 1$) whose spin and spatial symmetry
are symmetric. Since the Hamiltonian of the system depends only on the spatial
coordinate, there is no energy difference between fermionic and bosonic ground
states whose spatial symmetries are the same. Therefore, without taking into
account the symmetry of the variational quantum circuit, the quantum states
obtained by the VQE can be a mixture of fermionic and bosonic states, even
though an accurate energy value is obtained. We demonstrate such convergence
to a symmetry-neglected state in the numerical simulations.

The following describes the calculation conditions of the system and the
quantum circuits used for numerical simulations. Six qubits (5 qubits for
spatial coordinate and 1 qubit for spin coordinate) represent the one-body
space, and thus 12 qubits were used for the two-electron system. Spatial
coordinates of electrons $r_{1},r_{2}$ ranged from -0.5 bohrs to 0.5 bohrs and
the grid width $\varDelta r$ was $1/2^{5}$ bohrs = 0.03125 bohrs. $\epsilon$
was set to $\varDelta r/2$. The spatial grid employed in this study was
limited to a very coarse one, due to the computational resources of our
classical computer, but the realization of the FTQC is expected to allow
calculations with a grid fine enough to reproduce the electronic structure of
real materials. For variational circuits, the hardware efficient (HE) ansatz
[38] and the real-valued symmetry preserving ansatz (RSP) [37] were employed.
The HE ansatz consists of Ry gates and CNOT gates [39], which in the present
study had 6 layers.

(a)

(b)

(c)

Figure 4: Circuit architectures employed by VQE. (a) Symmetry-neglected (SN)
architecture; (b) Hartree-Fock (HF) architecture; (c) Multi-configuration (MC)
architecture. HE in the figure denotes the hardware efficient ansatz with 6
layers and RSP denotes the real-valued symmetry preserving ansatz.

In order to demonstrate the effects of the antisymmetrization and the multi-
configuration, we implemented the VQE with three different circuits. The
architectures of these circuits are shown in Fig. 4. The first architecture is
the symmetry-neglected (SN) architecture shown in Fig. 4(a), consisting of
consecutive HE ansatz blocks acting on the two-body space. In this
architecture, the antisymmetry of the seed state is not considered to be
preserved, and thus it is expected that the VQE will yield a neither
antisymmetric (fermionic) nor symmetric (bosonic) state. The number of blocks
of the HE ansatz was 6. The second architecture is the Hartree-Fock (HF)
architecture shown in Fig. 4(b), consisting of consecutive one-body space
variational circuits. In this architecture, the antisymmetry of the seed state
is preserved, but the VQE yields a ground state within the HF approximation.
The third architecture is the multi-configuration (MC) architecture shown in
Fig. 4(c), consisting of alternately layered one-body and two-body space
variational circuits. In this architecture, the VQE can achieve a ground state
beyond the HF approximation, and its energy is expected to be lower than that
obtained with the HF architecture.

To confirm this energy stabilization, potential energy curves were calculated
for the HF and MC architectures, at 16 points of interatomic distance
$|R_{1}-R_{2}|$ ranging from $\varDelta r$ to $16\varDelta r$ = 0.5 bohrs. The
number of one-body space variational circuits was 15 for both architectures
and that of two-body space variational circuits was 14 for the MC
architecture. For the optimization of variational parameters, the steepest
descent method with the Adam optimizer [40] was employed and the number of
optimization steps was set to 10000. All calculations for quantum circuits
were performed by a noiseless state vector simulator.

##  III Result and discussion

(a)

(b)

(c)

Figure 5: Many-body wave functions obtained by the VQE with (a) the SN
architecture, (b) the HF architecture and (c) the MC architecture at the
interatomic distance $|R_{1}-R_{2}|=$ 0.5 bohrs ($R_{1},R_{2}$ are shown by
pink dots).

###  III.1 Results of the VQE Calculations

Figure 5 shows the many-body wave functions obtained by the VQE with the
different architectures at an interatomic distance $|R_{1}-R_{2}|=$ 0.5 bohrs.
In the case of the wave function obtained with the SN architecture shown in
Fig. 5(a), the spatial coordinate is symmetric under the exchange of two
electrons, but its spin coordinate is neither antisymmetric nor symmetric
($\uparrow_{1}\uparrow_{2}+\uparrow_{1}\downarrow_{2}\neq\uparrow_{1}\uparrow_{2}+\downarrow_{1}\uparrow_{2}$).
This indicates that unless the symmetry for the variational quantum circuit is
not considered, the resulting state of the VQE can converge to such a
symmetry-neglected state. This is the notable difference from the second-
quantized VQE. In contrast, as shown in Figs. 5(b)(c), the wave functions
obtained with the HF and MC architectures are the antisymmetric singlet
states, as expected. The shape of the wave function obtained with the MC
architecture well reproduces that of the exact fermionic ground state shown in
Fig. 3, while that with the HF architecture is apparently different from it.
As will be described later, this difference in shape of the wave functions
indicates the difference in representability of the quantum states between the
HF approximation and CI theory.

The difference between them is also reflected in their ground state energies.
Figure 6(a) shows the potential energy curves obtained with the HF and MC
architectures. It can be seen that the ground state energies obtained with the
MC architecture reproduce the results of the exact diagonalization well,
whereas those obtained with the HF architecture converge to higher values.
This demonstrates the energy stabilization due to the multi-configuration
character, in other words, the effect of electron correlation, which is the
pillar of the quantum chemistry theories [41, 42].

(a)

(b)

Figure 6: (a) Potential energy curves and (b) entanglement entropy obtained by
exact diagonalization (black dashed lines) and VQE with the MC architecture
(blue solid lines) and the HF architecture (red solid lines). The calculated
potential energy curves are significantly different from those of the three-
dimensional hydrogen molecular system, which is attributed to the difference
in dimensions between the systems [43].

###  III.2 Analysis of Many-Body States based on Quantum Information Theory

Although the numerical experiments in this section confirmed that the proposed
method can reproduce the exact fermionic ground state well, the obtained many-
body wave function provides little insight into the microscopic electronic
structure. The electronic structure is mostly understood from the electron
orbital picture; however, since our method is based on a real-space basis, the
orbital picture of the obtained state is not obvious. Furthermore, a lack of
an orbital picture makes it impossible to quantitatively evaluate the multi-
configuration character of the obtained state. In order to get a clear orbital
picture and evaluate the multi-configuration character of obtained states, we
analyzed many-body quantum states from the perspective of quantum information
theory. Quantum information has been attracting increasing attention in recent
years as a key to understanding electron dynamics in chemical reactions and
strongly correlated materials [44, 45, 46, 47, 48, 49, 50, 51, 52, 53].

We will first quantify the multi-configuration character of many-body states.
To evaluate the multi-configuration character, we employed the entanglement
entropy, which is typically employed to measure the degree of entanglement
between subsystems. We shall now briefly explain the definition of
entanglement entropy. A many-body wave function of the two-electron system can
be decomposed into multiple product states by the Schmidt decomposition, as
follows:

$\ket{\psi(r_{1},r_{2})}=\sum_{i}\lambda_{i}\ket{\mu_{i}(r_{1})}\otimes\ket{\chi_{i}(r_{2})},$
(21)

where $\lambda_{i}$ are the Schmidt coefficients (superposition coefficients
of product states) and $\ket{\mu_{i}(r_{1})},\ket{\chi_{i}(r_{2})}$ are the
Schmidt basis on subsystems. The entanglement entropy $S$ is defined by the
Shannon entropy of the probability distribution $|\lambda_{i}|^{2}$ as

$S=-\sum_{i}|\lambda_{i}|^{2}\log_{2}|\lambda_{i}|^{2}.$ (22)

Considering the fact that the Shannon entropy of a normal distribution is
proportional to the logarithm of its variance, the larger the variance of
probability distribution $|\lambda_{i}|^{2}$, that is, the more product states
are superposed, the larger the entanglement entropy. Since the entanglement
entropy of a single Slater determinant
$\ket{\psi_{\rm{HF}}}=1/\sqrt{2}(\ket{\alpha}_{1}\otimes\ket{\beta}_{2}-\ket{\beta}_{1}\otimes\ket{\alpha}_{2})$
is 1, that of an MC state is expected to be larger than 1.

Figure 6(b) shows the entanglement entropies calculated for the ground states
obtained by the exact diagonalization and the MC architecture. As expected,
the entanglement entropy of the MC state is larger than 1 and increases as
interatomic distance increases, which is in good agreement with the exact
result. This behavior of entanglement entropy indicates that the ground state
becomes more multi-configurational as the interatomic distance approaches the
dissociation limit, which is a well-known fact in the field of conventional
quantum chemistry [41, 42].

We can now extract the electron orbital picture from the many-body state
represented in real space. From the expression of the Schmidt decomposition in
Eq. 21, Schmidt basis $\ket{\mu_{i}(r_{1})},\ket{\chi_{i}(r_{2})}$ can be
regarded as one-body wave functions, that is, the electron orbitals of the
electrons, and the Schmidt coefficients as their contributions to the many-
body state. Therefore, the Schmidt basis and the Schmidt coefficients provide
insight into the electron orbital picture of many-body wave functions.

Figure 7: Results of the Schmidt decomposition of
$\ket{\psi(r_{1\downarrow},r_{2\uparrow})}$. Left: First three Schmidt
coefficients $\lambda_{0},\lambda_{1},\lambda_{2}$ at each interatomic
distance. Right: Resulting product-state wave function at interatomic distance
$|R_{1}-R_{2}|$ = 0.5 bohrs ($R_{1},R_{2}$ are shown by pink dots), and
corresponding one-electron orbitals
$\ket{\mu_{i}(r_{1\downarrow})},\ket{\chi_{i}(r_{2\uparrow})}$ (black lines).
The one-electron orbitals are scaled for visibility.

Figure 7 shows the results of the Schmidt decomposition performed for one
anti-parallel spin subspace $\ket{\psi(r_{1\downarrow},r_{2\uparrow})}$ of the
exact ground state (upper left part of Fig. 3) and the MC state (Fig. 5(c)).
The left side of the figure shows the first three Schmidt coefficients
$\lambda_{0},\lambda_{1},\lambda_{2}$ of each interatomic distance. As shown,
the contribution of the zeroth product state $\lambda_{0}$ is nearly constant
at all interatomic distances, whereas that of the first product state
$\lambda_{1}$ increases as interatomic distance increases, which is consistent
with the behavior of the entanglement entropy shown in Fig. 6(b). This
suggests that the configuration interaction between the zeroth and first
product states, that is, their superposition, explains the energy
stabilization due to the multi-configuration character. The right side of the
figure shows the product states constituting the MC state at the interatomic
distance $|R_{1}-R_{2}|=$ 0.5 bohrs. The electron orbitals constituting each
product state are indicated by black lines in each figure. Using these
electron orbitals, we will illustrate the energy stabilization due to the
multi-configuration character in the following discussion.

The electron orbitals constituting the zeroth and first product states have
peaks at the proton positions $R_{1},R_{2}$ (pink dots in figure), and these
orbitals correspond to bonding and antibonding orbitals. Following the typical
explanation of the linear combination of atomic orbitals (LCAO) approximation,
let $\ket{p_{i}(r)}$ be the electron orbitals distributed around the $i$th
proton. Then the bonding and antibonding orbitals are represented as
$\ket{\psi_{\mathrm{\sigma}}(r)}=\ket{p_{1}(r)}+\ket{p_{2}(r)}$ and
$\ket{\psi_{\mathrm{\sigma^{*}}}(r)}=\ket{p_{1}(r)}-\ket{p_{2}(r)}$,
respectively. The bonding state of two electrons
$\ket{\psi_{\mathrm{\sigma\sigma}}(r_{1},r_{2})}$ is expressed as the tensor
product of the bonding orbitals $\ket{\psi_{\mathrm{\sigma}}(r)}$ as follows:

$\begin{split}&\ket{\psi_{\mathrm{\sigma\sigma}}(r_{1},r_{2})}\\\
&=\ket{\psi_{\mathrm{\sigma}}(r_{1})}\otimes\ket{\psi_{\mathrm{\sigma}}(r_{2})}\\\
&=\ket{p_{1}(r_{1})}\otimes\ket{p_{2}(r_{2})}+\ket{p_{2}(r_{1})}\otimes\ket{p_{1}(r_{2})}\\\
&\qquad+\ket{p_{1}(r_{1})}\otimes\ket{p_{1}(r_{2})}+\ket{p_{2}(r_{1})}\otimes\ket{p_{2}(r_{2})},\end{split}$
(23)

The four terms in the above expression correspond to the four peaks in the
zeroth product state and in the HF ground state shown in Fig. 5(b); thus,

$\ket{\psi_{\mathrm{\sigma\sigma}}(r_{1},r_{2})}\sim\ket{\mu_{0}(r_{1})}\otimes\ket{\chi_{0}(r_{2})}\sim\ket{\psi_{\mathrm{HF}}(r_{1},r_{2})}.$
(24)

The antibonding state, or two-electron excited state,
$\ket{\psi_{\mathrm{\sigma^{*}\sigma^{*}}}(r_{1},r_{2})}$ is expressed as the
tensor product of the antibonding orbitals
$\ket{\psi_{\mathrm{\sigma^{*}}}(r)}$ as follows:

$\begin{split}&\ket{\psi_{\mathrm{\sigma^{*}\sigma^{*}}}(r_{1},r_{2})}\\\
&=-\ket{\psi_{\mathrm{\sigma^{*}}}(r_{1})}\otimes\ket{\psi_{\mathrm{\sigma^{*}}}(r_{2})}\\\
&=\ket{p_{1}(r_{1})}\otimes\ket{p_{2}(r_{2})}+\ket{p_{2}(r_{1})}\otimes\ket{p_{1}(r_{2})}\\\
&\qquad-\ket{p_{1}(r_{1})}\otimes\ket{p_{1}(r_{2})}-\ket{p_{2}(r_{1})}\otimes\ket{p_{2}(r_{2})},\end{split}$
(25)

where $-1$ is taken as a global phase. The four terms in the above expression
correspond to the four peaks in the first product state; thus,

$\ket{\psi_{\mathrm{\sigma^{*}\sigma^{*}}}(r_{1},r_{2})}\sim\ket{\mu_{1}(r_{1})}\otimes\ket{\chi_{1}(r_{2})}.$
(26)

Superposing bonding and antibonding states equally yields

$\begin{split}&\ket{\psi_{\mathrm{\sigma\sigma+\sigma^{*}\sigma^{*}}}(r_{1},r_{2})}\\\
&=\frac{1}{2}\ket{\psi_{\mathrm{\sigma\sigma}}(r_{1},r_{2})}+\frac{1}{2}\ket{\psi_{\mathrm{\sigma^{*}\sigma^{*}}}(r_{1},r_{2})}\\\
&=\ket{p_{1}(r_{1})}\otimes\ket{p_{2}(r_{2})}+\ket{p_{2}(r_{1})}\otimes\ket{p_{1}(r_{2})}.\end{split}$
(27)

It can be seen that energetically unfavorable electron configurations
$\ket{p_{1}(r_{1})}\otimes\ket{p_{1}(r_{2})}$,
$\ket{p_{2}(r_{1})}\otimes\ket{p_{2}(r_{2})}$, where both electrons are
distributed around the same proton, are cancelled out and a lower energy state
is achieved. The two terms in the above expression correspond to the two peaks
in the MC state $\ket{\psi_{\mathrm{MC}}(r_{1},r_{2})}$ shown in Fig. 5(c);
thus,

$\ket{\psi_{\mathrm{\sigma\sigma+\sigma^{*}\sigma^{*}}}(r_{1},r_{2})}\sim\ket{\psi_{\mathrm{MC}}(r_{1},r_{2})}.$
(28)

This state represents the electron configuration where electrons are
distributed around different protons to avoid each other, reminiscent of
correlated, or entangled, electrons. To summarize, a multi-configuration state
is inherently an entangled state consisting of multiple product states, and
electron correlation can be interpreted as quantum entanglement in the
electron system.

Although this discussion on electron orbitals may be just a textbook subject
[41, 42], it is worth noting that such an orbital picture can be extracted
from a real-space represented by many-body states by using the Schmidt
decomposition. In addition, the minor orbital contributions, other than
bonding and antibonding orbitals, can be considered in our method. Indeed, as
shown in the left part of Fig. 7, the exact ground state and the MC state
include the contribution of the second product state
$\lambda_{2}\ket{\mu_{2}(r_{1\downarrow})}\otimes\ket{\chi_{2}(r_{2\uparrow})}$
whose electron orbitals have nodes at proton positions and are distributed in
the interstitial region. Since a real-space basis is a complete basis set of a
discrete space, it can represent both localized atomic orbitals and such
delocalized orbitals, freeing us from the need for careful selection of basis
functions in typical quantum chemical calculations. From this point, a real-
space basis is an attractive option for the quantum simulation in the FTQC
era.

The results shown in this section demonstrate one possible framework of
quantum chemistry in the FTQC era, where we understand electronic structures
based on electron orbitals derived from many-body wave functions represented
in real space. In this study, many-body wave functions were analyzed by a
classical computer; however, since it is almost impossible to obtain a state
vector of even a quantum circuit of a few dozen qubits, this analysis must be
performed on a quantum computer. Fortunately, several methods have been
proposed to perform singular value decomposition on quantum computers [54, 55,
56], which may allow such analysis to be performed on a quantum computer with
reasonable computational resources.

The problem concerned in our method is the barren plateaus [57, 58, 59] in the
optimization process of the variational state. In fact, the disappearance of
gradients of variational parameters was observed during the optimization
process, which required a large number of optimization steps. Such barren
plateaus are the most serious problem in most variational quantum algorithms.
Nevertheless, as mentioned in the Introduction, since by using the QPE, the
true ground state can be distilled from the approximated ground state prepared
by the VQE, the incomplete convergence of the VQE due to the barren plateaus
will be tolerated to some extent. The complementary combination of the VQE and
the QPE will allow the state preparation for first-quantized methods.

##  IV Summary and Conclusion

In this paper, we propose a state preparation method for the first-quantized
quantum simulations based on the real-space representation. We employ a
variational method for preparing ground states, and provide a design principle
for constructing antisymmetrized variational quantum circuits. Our proposed
circuits are capable of generating a superposition of exponentially large
number of Slater determinants, that is, MC states with polynomial numbers of
quantum gates. We performed VQE calculations for a 1D-$\rm{H_{2}}$ system and
confirmed that the proposed circuit reproduces well the exact fermionic ground
state. In addition to performing VQE, we quantitatively evaluated the multi-
configuration character of many-body states by using the entanglement entropy
between two electrons, and extracted the electronic orbital picture from many-
body states represented in real space by the Schmidt decomposition.

Quantum computers, as demonstrated in this study, have great potential to
simulate many-body electron systems and shed light on their quantum
information. We believe that our proposed method will contribute to realizing
the first-quantized simulation for electron dynamics and will bring a deeper
understanding of electron systems to materials science in the FTQC era.

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