# Effect of constraint and Tabu Search term on Variational Quantum Eigensolver
and Subspace-Search Variational Quantum Eigensolver

Hikaru Wakaura<EMAIL_ADDRESS>QuantScape Inc. 4-11-18, Manshon-
Shimizudai, Meguro, Tokyo, 153-0064, Japan  Takao Tomono
<EMAIL_ADDRESS>Digital Innovation Div. Toppan Printing Co. Ltd., 1-5-1,
Taito, Taito, Tokyo, 110-8560, Japan

(23 March 2021)

###### Abstract

Subspace-Search Variational Quantum Eigensolver(SSVQE) is searching method of
multiple states and relies on the unitarity of transformations to ensure the
orthogonality of output states for multiple states. Therefore, this method is
thought to be promising method for quantum chemistry because ordinary
Variational Quantum Eigensolver (VQE) can only calculate the excited states
step by step from ground state based on Variational Quantum deflation (VQD).
We compare the advantage of VQE, SSVQE with/without the constraint term and/or
Tabu search term, that are added by Lagrange’s multiplier method so as to
calculate the desired energy levels. We evaluated the advantage by calculating
each energy levels of H2 and HeH, respectively. As there simulation results,
the accuracy calculated by constrained VQE with Tabu search indicates higher
accuracy than that of our other algorithm, for analysis on H2. The accuracy
calculated by constrained SSVQE indicate higher that of the constrained VQE
with Tabu search. We found it is beneficial for enhance the accuracy to use
constraint terms decreasing convergence times to use Tabu search terms
according to the nature of molecules. We demonstrate that constraint and Tabu
search terms contribute to the accuracy and convergence time on quantum
chemical calculating.

Keywords: VQE ; SSVQE ; constraint term ; Tabu search term ; quantum chemistry

###### Contents

  1. I Introduction
  2. II Methods
    1. II.1 Variational Quantum Eigensolver (VQE) method
    2. II.2 Subspace Search VQE
    3. II.3 Addition to Constraint and Tabu Search
    4. II.4 Preparation of calculation
  3. III Numerical Simulation
    1. III.1 The effect of Constrained and Tabu Search term on calculation of H2
    2. III.2 The effect of constrained and Tabu Search term on calculation of HeH
  4. IV Discussion
  5. V Concluding Remarks

##  I Introduction

Quantum computer is attractive equipment because the principle is based on
quantum mechanics. Quantum computers with a hundred qubits will be develop as
quantum computer with 54 qubits that reaches quantum supremacy is announced in
2019[1]. Though it takes much time to get perfect quantum computer with more
than 1-milion qubits, near-term quantum computers are used as noisy
intermediate-scale quantum (NISQ) devices. The device is hybrid quantum
computer that consist of classic and quantum computers, currently. Within the
device, quantum information treatment is done within coherence time on quantum
computer, currently. Reflecting the fact that present quantum computers are
not fault tolerant, they do not have a guaranteed accuracy of the
computational result[2]. However, such a NISQ device is believed to be
considerably attractive if the gate fidelity is sufficiently high. This fact
encourages us to look for practical applications of them.

The VQE is used as one of variational quantum algorithm used in NISQ device.
The algorithm is to find an approximate ground state of a given Hamiltonian H.
The group read by Dr. Aran Aspuru-Guzik developed the VQE in 2011[3]. This
approach uses a classical optimization routine to minimize the expected energy
of candidate wave functions, using the quantum device to evaluate the
predicted energy[4]. Essentially, the VQE leverages the unique capacity of
quantum circuits to prepare entangled states from classical sampling.
Essential ingredients of the VQE algorithm have recently been demonstrated on
a variety of experimental platforms[5][6][7]. These initial experiments
indicate a robustness to systematic control errors (so-called coherent errors)
which would preclude fully quantum calculations, as well as a possibility of
scaleout of quantum circuit depth with Hamiltonian complexity.

To expand the potential application of the VQE on the ground state, many
researches have extended the method to evaluate properties of excited states
of a target Hamiltonian. Several of algorithms among such extensions are the
subspace-search VQE (SSVQE)[8], the multi-state contracted VQE (MCVQE)[9], and
the variational quantum deflation (VQD)[10]. The SSVQE and the MCVQE can
readily evaluate the transition amplitude[11]. Besides, other algorithms to
calculate the energy of molecules have been proposed[12][13][14]. On the other
hands, we have limitation of application of VQD method since the transition
amplitude is related to properties of the system such as the absorption or
emission spectrum of photon. The SSVQE is the method that takes two or more
orthogonal states as inputs to a parametrized quantum circuit, and minimizes
the expectation value of the energy in the space spanned by those states. This
method automatically generates the orthogonal condition on the output states,
and is possible for us to remove the swap test[15], which has been employed in
the previous works to ensure the orthogonality.

Constrained Algorithm is introduced to Loop Quantum Gravity system as Master
constrained algorithm [16]. This constrained condition is developed for
overcoming the complications associated with Hamiltonian constrain of the
Dirac algebra. We can control the complications by moving of spin pair in
quantum algorithm. These are three ’no-go’s’ that are all well-known
consequences of standard nonrelativistic Hilbert space quantum theory.
However, like Einstein’s radical re-derivation of Lorentz’s transformation
based upon privileging a few simple principles, we here introduce the above
constraints term to the level of fundamental information-theoretic ’laws of
nature’ from which quantum theory can, we claim, be deduced.

The tabu search (TS) algorithm[17] are one of the meta-heuristic search
algorithms devised by Fred Glover. Therefore, it is simply implemented method
to iteratively find a near-optimal solution, and it has been successfully used
to solve various optimization problems. For example, optimization problem of
network used for telecommunication can be solved by TS with Quantum
Approximate Optimization Algorithm(QAOA) efficiently[18]. The TS algorithm
with a flexible memory system has the ability to climb out of local minima,
suffers from the tuning of the tabu tenure, meaning that it still becomes
stuck at local minima and has a low speed of convergence. It takes a great
deal of time to escape to near-global optimal from current position.
Therefore, both intensification and diversification strategies should be
considered to improve the robustness, effectiveness and efficiency of simple
TS; a more powerful neighborhood structure can be feasibly constructed by
applying quantum computing concepts.

In this paper, we compared to effect of complex complex complex algorithm that
are VQE, SSVQE with/without constraint term and/or tabu search term for
molecules of H2 and HeH. Simulation methods are introduced firstly. Next,
initialization will be explained in detail. After these simulation, we denote
the simulation results and discuss about utilization of these methods based on
the results. We conclude that constraint and Tabu search terms are beneficial.

##  II Methods

In this section we describe VQE and SSVQE with/without constraint term and /or
tabu search term. Firstly, we explain how to find the minimum value of excited
energy levels by VQE method and its flowchart with quantum circuit to perform
it including equation. Secondary, we explain about SSVQE method.

###  II.1 Variational Quantum Eigensolver (VQE) method

A flowchart outline of the VQE algorithm is shown in Fig. 1. We must prepare
initial parameter set ${\bm{\theta}}_{i=0}$ on classical computer and then we
calculate eigenvalue $E({\bm{\theta}}_{i})$ of energy by using ansatz on
quantum circuit, and then the classical computer decide whether or not the
eigenvalue $E({\bm{\theta}}_{i+1})$ is smaller than settled minimum eigenvalue
$E_{0}$. If the eigenvalue $E({\bm{\theta}}_{i+1})$ is not smaller than
minimum eigenvalue $E_{0}$, optimization of parameter set ${\bm{\theta}}$ is
done by using Powell method on classical computer. After optimization, new
eigenvalue $E({\bm{\theta}}_{i+1})$ is substituted for ansatz in the quantum
circuit instead of $E({\bm{\theta}}_{i})$. When a determination condition
$(E({\bm{\theta}}_{i+1})=E_{0})$ is not satisfied, the above processing is
repeated until the determination condition is satisfied. An quantum algorithm
is presented by second-order quantum hamiltonian $H$ as follows.

$H=\sum_{j,k=0}h_{jk}c_{j}^{\dagger}c_{k}+\sum_{j,k,l,m=0}\langle jk\mid\mid
lm\rangle c_{j}^{\dagger}c_{k}^{\dagger}c_{l}c_{m}.$ (1)

To calculate the matrix elements $E$, we use the quantum circuit to evaluate
the inner products,

$E=\langle\Phi\mid H\mid\Phi\rangle,$ (2)

where $\Phi$ is taken to be the initial approximate ground state $\Phi$.
Optimization is performed by solving ground state $\Phi$ to minimize $E_{0}$.
The state is represented by Slater’s determinant represented by occupied
and/or unoccupied orbitals. For example, we consider the case of 4-qubits
system. Each one qubit is represent as $\mid 0\rangle$ and $\mid 1\rangle$.
One state of the ground slater determinant is represented as $\mid
1100\rangle$ because there are two occupied bonding orbitals and two
unoccupied anti-bonding orbitals. Here, 1 indicates the orbital is occupied
and 0 indicates the orbital is unoccupied, respectively. An left hand side 11
of $\mid 1100\rangle$ is bonding orbitals and right hand side of that is anti-
bonding orbitals. Each term of hamiltonian eq.1 is one-body integral and two-
body integral, respectively. Indices of them indicate the index of the
orbital. In general, any materials are stable when their energy level is
lowest. Comparing the energy of bonding orbital to anti-bonding orbital,
bonding orbital is more stable. hydrogen molecules are in the ground state
when their two electrons are both in the bonding orbitals as up-spin and down-
spin pair. Bonding orbitals are referred to $\sigma$ bonding orbitals. The
bonding orbital is expressed for the basis function of STO-3G. All terms of
UCC and hamiltonian are expressed as Pauli operator by Jordan-Wigner or
Bravyi-Kitaev transformation[19, 20]. A transformed hamiltonian is represented
by

$\displaystyle H$ $\displaystyle=$ $\displaystyle
f_{0}1+f_{1}\sigma^{z}_{0}+f_{2}\sigma^{z}_{1}+f_{3}\sigma^{z}_{2}+f_{1}\sigma^{z}_{0}\sigma^{z}_{1}$
$\displaystyle+$ $\displaystyle
f_{4}\sigma^{z}_{0}\sigma^{z}_{2}+f_{5}\sigma^{z}_{1}\sigma^{z}_{3}$
$\displaystyle+$ $\displaystyle
f_{6}\sigma^{x}_{0}\sigma^{z}_{1}\sigma^{x}_{2}+f_{6}\sigma^{y}_{0}\sigma^{z}_{1}\sigma^{y}_{2}$
$\displaystyle+$ $\displaystyle
f_{7}\sigma^{z}_{0}\sigma^{z}_{1}\sigma^{z}_{2}+f_{4}\sigma^{z}_{0}\sigma^{z}_{2}\sigma^{z}_{3}+f_{3}\sigma^{z}_{1}\sigma^{z}_{2}\sigma^{z}_{3}$
$\displaystyle+$ $\displaystyle
f_{6}\sigma^{x}_{0}\sigma^{z}_{1}\sigma^{x}_{2}\sigma^{z}_{3}+f_{6}\sigma^{y}_{0}\sigma^{z}_{1}\sigma^{y}_{2}\sigma^{z}_{3}+f_{7}\sigma^{z}_{0}\sigma^{z}_{1}\sigma^{z}_{2}\sigma^{z}_{3}.$

Generally speaking, we compute excited-states energy after calculating ground-
state energy on quantum chemistry calculation. We prepare cluster terms T to
carry out transition from a ground state to an excited state as we can not
calculate wave function on excited-state by only hamiltonian. The cluster is
called as Unitary Coupled Cluster (UCC). We apply Unitary Coupled Cluster of
Single and Double (UCCSD) on the condition of single and double excitation
terms[21]. Cluster terms T is expressed as,

$T=\sum_{j\in occu.,k\in vac.}\theta_{k}^{j}c_{j}^{\dagger}c_{k}+\sum_{j,k\in
occu.,l,m\in vac.}\theta_{lm}^{kj}c_{j}^{\dagger}c_{k}^{\dagger}c_{l}c_{m}.$
(4)

We introduce ansatz on quantum circuit to multiply Hamiltonian by
$exp(i(T-T^{\dagger}))$ for excited state energy. After that, Hamiltonian and
Cluster terms are decomposed by Suzuki-Trotter transformation [22]. As the
depth of circuit (repeating ratio) increases, the variable coefficients become
smaller. If the depth of circuit is much larger than 100, we will not need
variable coefficient. In the case, calculation time become infinite.
Therefore, the depth had better be set two. As preparation of optimization, we
seek $\theta_{k}$ of Pauli operator by using $exp(-i\theta_{k}P_{j}t)$ on
quantum computer. $\theta_{k}$ is variable coefficient of k-th term, $P_{j}$
is j-th Pauli operator, and t is the coefficient. We can prepare to make
excited state energy.

Figure 1: Flowchart of VQE method. Purple area indicates the processes
performed by quantum computer and yellow area indicates the processes
performed by classical computer, respectively. Quantum computer calculate the
energy levels, constraint terms, Tabu search, and deflation terms. Classical
computer optimizes the parameter set $\bm{\theta}$ by Powell method[23].
Figure 2: Quantum circuit to perform the operation that act on
$exp(-i\theta\sigma_{0}^{\alpha}\sigma_{1}^{\beta}\sigma_{2}^{\gamma}\sigma_{3}^{\delta})$.
$\alpha$, $\beta$, $\gamma$, and $\delta$ determine the gate $G_{j}$ and
$G_{j}^{\dagger}$ right and left of $q_{0}$, $q_{1}$, $q_{2}$, and $q_{3}$
corresponding gate for $x$, $y$, $z$ in the list, respectively.

The quantum circuit to solve $exp(-i\theta_{k}P_{j}t)$ by Hamiltonian is as
shown in Fig. 2. This quantum circuit is four-qubits system. This quantum
circuit acts on multiple term of pauli operater. $q_{0}-q_{3}$ indicate first,
second, third, and fourth qubit, respectively. $q_{0}$ and $q_{1}$ correspond
to bonding orbitals and $q_{2}$ and $q_{3}$ correspond to anti-bonding
orbitals. $R_{z}(\theta_{k})$ corresponds to the variable coefficient in eq.4.
$G_{j}$ and $G^{\dagger}_{i}$ are eigenvalue-operator and its conjugate
operator of pauli operator acted on i-th qubit, respectively. For example,
$G_{0}$ become H gate, $G_{1}$ become $R_{x}(\pi/2)$ gate and $G_{2}$ and
$G_{3}$ become I gate if we multiply Hamiltonian by
$exp(-i\theta\sigma_{0}^{x}\sigma_{1}^{y}\sigma_{2}^{z})$. Then, there are not
CNOT gate between $q_{2}$ and $q_{3}$. When we multiply Hamiltonian by
$exp(-i\theta\sigma_{0}^{x}\sigma_{2}^{y}\sigma_{3}^{z})$, CNOT gate move from
between $q_{0}$ and $q_{1}$ to $q_{0}$ and $q_{2}$. And, there is no CNOT gate
between $q_{1}$ and $q_{2}$. $G_{1}$ disappear, $G_{2}$ becomes $R_{x}(\pi/2)$
and $G_{3}$ becomes $I$ gate. By using VQE method we repeat the try and error
by acting on the hamiltonian and cluster on quantum computers and change the
variable coefficient to optimize in this way. The evaluation function of i-th
state is represented as,

$\displaystyle F_{i}(\bm{\theta})$ $\displaystyle=$
$\displaystyle\langle\Phi_{ini}\mid
UHU^{\dagger}\mid\Phi_{ini}\rangle+E_{i}^{def.}.$ (5)

Then,

$U=\Pi_{j,k}exp(i\theta_{k}P_{j}t_{j}).$ (6)

$\mid\Phi_{ini}\rangle$ is the ground state $\mid 1000\rangle$ of the system
transformed by Bravyi-Kitaev method. $E_{i}^{def.}$ and indicate the deflation
of i-th state. Deflation in eq.5 become zero when the value of energy is
optimized. However, we need to seek the value of energy on excited state.
Deflation term is necessary to derive excited states. Deflation term is so
called Variational Quantum Deflation (VQD) method. VQD method is performed by
adding overlap of previous and present states to evaluate wave function. The
form is expressed by

$E(\bm{\theta})=\langle\Phi_{i}\mid(H+A\sum_{j<i}\mid\Phi_{j}\rangle\langle\Phi_{j}\mid)\mid\Phi_{i}\rangle.$
(7)

Here, A is coefficient to weigh. The overlap of i and j state can be derived
by SWAP-test algorithm.

###  II.2 Subspace Search VQE

Subspace-Search Variational Quantum Eigensolver (SSVQE) method is advanced
algorithm of VQE method that can drive multiple states at once. As described
in introduction, this method automatically generates the orthogonal condition
on the output states. The method is optimization method of energy for not each
state but summation. The summation of energy is represented by

$\displaystyle F(\bm{\theta})$ $\displaystyle=$
$\displaystyle\sum_{j}^{N_{d}}w_{j}\langle\Phi_{j}\mid
H\mid\Phi_{j}\rangle+\sum_{j}^{N_{d}}E_{j}^{def.}$ $\displaystyle=$
$\displaystyle\sum_{j}^{N_{d}}E_{j}+\sum_{j}^{N_{d}}E_{j}^{def.}$
$\displaystyle When~{}i$ $\displaystyle>$ $\displaystyle j,~{}E_{i}>E_{j}.$

Here, we calculate for solving minimum energy on the condition of $i>j$,
$w_{i}<w_{j}$. $w_{i}$ and $w_{j}$ are weight constraints.$N_{d}$ is the
number of states that are driven at once. Besides, each initial state of
$\mid\Phi_{j}\rangle$ is different for j. All the cluster should be common on
all calculation states. There is some possibility of having low accuracy when
excited state cross each other. Therefore, in this time, we calculate all
clusters for each state to keep high accuracy without being trapped by local
minimums. Deflation term is set to be A = 1 for all previous states. Excited
states can be derived spontaneously in the order of $i>j$, $E_{i}>E_{j}$ when
deflation term is zero.

###  II.3 Addition to Constraint and Tabu Search

$E_{i}^{const}$ indicate Constraint term of i-th state. Constraint terms can
be derived by Lagrange’s multiplier method[24] into eq.5. The constraint
condition of i-th state $E_{i}^{const}$ is represented by

$E_{i}^{const}=\sum_{j=0}^{num~{}of~{}const.}\mid\langle\Phi_{i}\mid(U_{j}-U_{j}^{const})\mid\Phi_{i}\rangle\mid.$
(10)

$U_{j}$ indicates the observable as constraint term and $U_{j}^{const}$ is
targeted value. Here, $U_{j}$ include $\sum s_{i}^{2}$, $\sum s_{i}^{z}$ and
$N$. Then, the s is spin parameter including magnetic moment that we can
observe as constraints and $N$ is the number of electrons. This term is
calculated in the same way as for hamiltonian. We have many local minimums
look like Rastregin function when energy lift up to excited state. As the
result, electron tend to catch on the local minimums when electron drop in
ground state. Therefore, we use Powell method as newtonian method is tend to
be trapped on them. We add constraint term and Tabu Search terms on excited
state. Firstly, Tabu search term is expressed by

$\displaystyle E^{Tabu}$ $\displaystyle=$
$\displaystyle\sum_{j}^{num.~{}of~{}Tabu.}\exp{(-\mu\langle\Phi_{j}\mid(U_{j}-U_{j}^{Tabu.})\mid\Phi_{j}\rangle^{2})}a$
$\displaystyle=$ $\displaystyle\sum_{j}^{num.~{}of~{}Tabu.}E_{j}^{Tabu.}.$

This Tabu Search term is very useful to avoid being trapped by local minimums.
Here, $\mu$ is the width and a is the amplitude of tabu Search term in
parameter space, respectively. $U_{j}^{Tabu.}$ indicates the value that must
be avoided for $U_{j}$. If we use the system with degeneration in SSVQE
method, the solution that is satisfied with eq.9 emerge according to the
number of degeneracy. To avoid these solutions, The evaluation function is set
by,

$F_{j}(\bm{\theta})=\langle\Phi_{j}\mid
H\mid\Phi_{j}\rangle+E^{const.}_{j}+E^{Tabu}_{j}.$ (12)

This evaluation function is satisfied with
$F_{i}(\bm{\theta})>F_{j}(\bm{\theta})$ when i is larger than j. We simulate
the energy levels of ground state, excited state (triplet, singlet, and
doubly) by using VQE and SSQVE with with Constraint and/or Tabu Search terms.
We simulate the initial states on calculation on H2 as $\mid 1000\rangle$,
$\mid 0110\rangle$, $\mid 1100\rangle$, and $\mid 0010\rangle$, respectively.
And, we solve initial state according to every two levels. Tabu terms are
fixed to avoid $\sum s_{i}^{2}=0.75$, $\sum s_{i}^{z}=10000$, and $N=10000$,
respectively. In the subsection 3.2, we simulate the initial states on
calculation on HeH as $\mid 1110\rangle$, $\mid 1101\rangle$, $\mid
1011\rangle$, and $\mid 0111\rangle$. And, we solve initial state according to
every two levels too. Tabu terms are fixed to avoid $N=10000$.

###  II.4 Preparation of calculation

In this time, we use Powell method as classical algorithm. Thenthe number of
iterations is limited in 2000 times at a maximum. Here, $\mu$ is 100 and a is
100 for all evaluation functions. Deflation terms $E^{def.}_{i}$ and
constraint term are as follows. Deflation term of i state $E^{def.}_{i}$ is
expressed by

$\displaystyle E_{i}^{def.}$ $\displaystyle=$ $\displaystyle((af+b(1-f))$
$\displaystyle\times$ $\displaystyle(\sum_{j<i}(exp(r-0.25r_{d})+1)^{-1}$
$\displaystyle\times$
$\displaystyle\mid\langle\Phi_{j}\mid\Phi_{i}\rangle\mid{{}^{2}}$
$\displaystyle+$ $\displaystyle(1-(exp(r-0.25r_{d})+1)^{-1})$
$\displaystyle\times$ $\displaystyle
f(\mid\langle\Phi_{j}\mid\Phi_{i}\rangle\mid{{}^{2}})).$

a and b, and $f=(exp(\alpha(r-rd))+1)^{-1}$ indicate two constants and
diatomic bond length r of Fermi-Dirac distribution respectively. Then, $r_{d}$
is a given diatomic bond length. a is 1.0 and $\alpha$ is 100.
$f(\mid\langle\Phi_{j}\mid\Phi_{i}\rangle\mid^{2})$ is inhomogenius function
of overlap of i and j states to derive degenerated excited states and is
expressed by

$\displaystyle f(\mid\langle\Phi_{j}\mid\Phi_{i}\rangle\mid{{}^{2}})$
$\displaystyle=$
$\displaystyle(1+2(\sqrt{5}+1))r^{4}/r_{d}^{4}E_{p}(r)/4\mid\langle\Phi_{j}\mid\Phi_{i}\rangle\mid{{}^{4}}$
$\displaystyle+$ $\displaystyle
2(\sqrt{5}+1)r^{4}/r_{d}^{4}E_{p}(r)/4\mid\langle\Phi_{j}\mid\Phi_{i}\rangle\mid{{}^{2}}.$

$E_{p}(r)$ is the value of one lower energy level for given r. All
calculations are performed numerically using blueqat SDK[25].

##  III Numerical Simulation

In this section, we simulate the result of ground and excited states on H2 and
HeH. We compared constraints with Tabu Search on VQE and SSVQE. We obtain the
value of energy state and accuracy of the value. Where, we obtained the
relationship between diatomic bond length ($\AA$) vs energy value (Hartree) on
each states by simulation. We obtained the relationship between diatomic bond
length($\AA$) vs accuracy data. Here, we use $Log_{10}(E-E_{FCI})$ as the
indicator of accuracy. FCI means Full-CI calculation based on Classic
algorithm.

###  III.1 The effect of Constrained and Tabu Search term on calculation of H2

We show the result of calculation of energy for diatomic bond length on H2 by
(1) VQE, (2) constrained VQE, and (3) constrained VQE with tabu Search terms
in Fig.3 (A), (B), and (C). Moreover, we show the corresponding accuracy of
calculation for diatomic bond length in Fig.4 (A), (B), and (C). In the case
of (1), the error bars of energy calculation increase except for ground state
as the distance between hydrogen bond become small as shown in Fig. 3 (A). The
error bars of energy on calculation of constrained VQE with/without Tabu
Search become smaller compared to the calculation on VQE, as shown in Fig.3
(B), and (C). However, we cannot recognize the effect on Tabu Search term when
the error bar of energy on excited state become larger as shown in Fig.3 (B),
and (C). We compared the accuracy of each data on VQE method as shown in Fig.4
(A), (B), and (C). The accuracy data of ground state on VQE (Fig.4 (1)) is
almost same as that on constrained VQE with/without Tabu Search (Fig.4 (B),
and (C)). The accuracy data on ground state are more or less below negative
forth power as average as shown in Fig.4 (B). The accuracy data of singlet and
doubly excited state on VQE is smallest than other states as shown in Fig.4
(A), (B), and (C). The accuracy data of ground state on constrained VQE is
smaller than that on conventional VQE method. We cannot recognized the effect
of Tabu Search (Fig.4 (B), and (C)).

Next we compared the effect of constrained and Tabu Search in the case of
SSVQE. We show the result of calculation of energy for diatomic bond length on
H2 by (4) SSVQE, (5) constrained SSVQE, and (6) constrained SSVQE with Tabu
Search in Fig.3 (D), (E), and (F). Moreover, we show the corresponding
accuracy of calculation for diatomic bond length in Fig.4 (D), (E), and (F).
We confirmed the error bar of ground state energy as shown in Fig.3 (D).
However, the error bar on constrained SSVQE with/without Tabu search became
small compared with that on SSVQE. The accuracy data of ground and triplet
states on constrained SSVQE with/without Tabu Search is about negative second
power on simulated all the range though that on SSVQE is about negative first
power. Therefore, we cannot recognize the effect of SSVQE.

From these results, we will select constrained VQE with/without Tabu search
for solution of H2. To investigate the effect of Tabu search, we analyze the
convergence of energy level as shown in Fig. 5. (a), (b), (c), and (d) shows
convergence results in the case of (2), (3), (5), and (6).Tabu search
contribute to stabilize the convergence of energy levels. As shown in Fig. 5
(a), (b), singlet and doubly excited states in the case of (2) spike 6 times
at most before convergence about in 2100 updates of variables. In contrast,
these two states in the case of (3) spike only 5 times at most before
convergence in about 1800 updates of variables.

Moreover, SSVQE method Tabu search contribute to stabilize the convergence
more than VQE method with Tabu search. As shown in Fig. 5(c),(d), there are
spikes of 11 times in the case of (5) before convergence in 4438 times. In
contrast, two states in the case of (6) spike only 3 times before convergence
in about 1600 updates of variables. Besides, the improvement of the accuracy
of these two states in case of (6) compared to (5) is greater than that of (3)
compared to (2). On these VQE and SSVQE, introducing Tabu search contributes
to decrease convergence times on calculation on H2.

Figure 3: Diatomic bond length($\AA$) of hydrogen molecule v.s.the energy
levels (Hartree) of each state calculated by VQE method of the case (1), (2),
(3), (4), (5), and (6). Solid line on each state is connecting average points
by ten sampling data. (A) Case (1): calculated by VQE method. (B) Case (2):
calculated by constrained VQE method. (C) Case (3): calculated by constrained
VQE method with Tabu search. (D) Case (4): calculated by SSVQE method. (E)
Case (5): calculated by constrained SSVQE method. (F) Case (6): calculated by
constrained SSVQE method with Tabu search.

Figure 4: Diatomic bond length($\AA$) of hydrogen molecule v.s. the accuracy
of calculated energy levels ($log_{10}(\mid E-E_{FCI}\mid)$) by VQE method of
the case (1), (2), (3), (4), (5), and (6). Solid line on each state is
connecting average points by ten sampling data. (A) Case (1): calculated by
VQE method. (B) Case (2): calculated by constrained VQE method. (C) Case (3):
calculated by constrained VQE method with Tabu search. (D) Case (4):
calculated by SSVQE method. (E) Case (5): calculated by constrained SSVQE
method. (F) Case (6): calculated by constrained SSVQE method with Tabu search.

Figure 5: Number of updates of variables v.s. Energy difference of singlet and
doubly excited states of H2 molecules between their grobal minimums,
respectively for the case of (2), (3), (5), (6), respectively. (a) number of
iteration v.s. Energy difference in the case of (2): constrained VQE. (b)
number of iteration v.s. Energy difference in the case of (3): constreained
VQE with Tabu search. (c) number of iteration v. s. energy difference in the
case of (5): constrained SSVQE. (d) number of iteration v.s. Energy difference
in the case of (6): constrained SSVQE with Tabu search.

###  III.2 The effect of constrained and Tabu Search term on calculation of
HeH

We show the result of calculation of energy for diatomic bond length on HeH by
(7) VQE, (8) constrained VQE, and (9) constrained VQE with Tabu Search terms
in Fig.6 (A), (B), and (C). Moreover, we show the corresponding accuracy of
calculation for diatomic bond length in Fig.7 (A), (B), and (C). In the case
of (7), (8) and (9), errors bar on excited states became huge large though we
could not find errors on ground states 1. The behavior (-4 as log) of ground
state 1 is almost same as that on H2 on VQE method. On the other excited
states, the accuracy is about 0 to -2.

We show the result of calculation of energy for diatomic bond length on HeH by
(10) SSVQE, (11) constrained SSVQE and (12) constrained SSVQE with tabu Search
terms in Fig.6 (D), (E), and (F). Moreover, we show the corresponding accuracy
of calculation for diatomic bond length in Fig.7 (D), (E), and (F). We cannot
find error bar on each energy states when we use the method of SSVQE. When
diatomic bond length is beyond 0.5 $\AA$, accuracy became below negative third
power on ground and excited 2 states. Moreover, the accuracy data on
constrained SSVQE with/without Tabu Search became negative third power to
negative forth power on all the range beyond 0.5 $\AA$. Here, we could not
recognize the difference with/without Tabu Search only for results.

As described above, energy error obtained by SSVQE is smaller than that by VQE
for HeH. To investigate convergence condition, we analyze the convergence of
energy of all the methods. Fig. 8 shows energy difference vs iteration in the
case of (7), (8), (9), (10), (11) and (12). Fig. 8 (a) denotes VQE method, (b)
constrained VQE method, and (c) constrained VQE method with Tabu Search, for
HeH, respectively. Fig. (d) denotes SSVQE, (e) constrained SSVQE method, and
(f) constrained SSVQE with Tabu Search, for HeH, respectively. As shown in
Fig. 8 (a), (b), (c), doublet excited states in the case of (7) and (8) spike
3 times before convergence in 1400 updates of variables. In contrast, these
states in the case of (9) spike only 2 times before convergence in about 900
updates of variables. Besides, the energy levels of these two states approach
global minimum, much faster than those of case (8). Tabu search term
contribute to stabilize the convergence on VQE and SSVQE methods for analysis
of energy level of HeH. As shown in Fig. 8 (d), (e), (f), especially those
energy levels in the case of (10) and (11) spike 3 times before convergence in
about 1500 updates of variables. In contrast, those two states in the case of
(12) spike only 2 times before convergence in about 1000 updates of variables.
Besides, one of these state (E2) approach global minimum twice faster than
that in the case of (11). Intensity of spikes is weaker than that in the case
of (11) too. On these VQE and SSVQE, introducing Tabu search contributes to
decrease convergence times on calculation on HeH too.

Figure 6: Diatomic bond length($\AA$) of heliuym hydrite v.s.the energy levels
(Hartree) of each state calculated by VQE method of the case (7), (8), (9),
(10), (11), and (12). Solid line on each state is connecting average points by
ten sampling data. (A) Case (7): calculated by VQE method. (B) Case (8):
calculated by constrained VQE method. (C) Case (9): calculated by constrained
VQE method with Tabu search. (D) Case (10): calculated by SSVQE method. (E)
Case (11): calculated by constrained SSVQE method. (F) Case (12): calculated
by constrained SSVQE method with Tabu search.

Figure 7: Diatomic bond length($\AA$) of heliuym hydrite v.s. the accuracy of
calculated energy levels ($log_{10}(\mid E-E_{FCI}\mid)$) by VQE method of the
case (7), (8), (9), (10), (11), and (12). Solid line on each state is
connecting average points by ten sampling data. (A) Case (7): calculated by
VQE method. (B) Case (8): calculated by constrained VQE method. (C) Case (9):
calculated by constrained VQE method with Tabu search. (D) Case (10):
calculated by SSVQE method. (E) Case (11): calculated by constrained SSVQE
method. (F) Case (12): calculated by constrained SSVQE method with Tabu
search.

Figure 8: Number of updates of parameters v.s. Energy difference of doublet
excited states of HeH molecules between their grobal minimums, respectively
for the case of (7), (8), (9), (10), (11), and (12), respectively. (a) number
of iteration v.s. Energy difference in the case of (7): VQE method. (b) number
of iteration v.s. Energy difference in the case of (8): constrained VQE. (c)
number of iteration v.s. Energy difference in the case of (9): constreained
VQE with Tabu search. (d) number of iteration v.s. Energy difference in the
case of (10): SSVQE method. (e) number of iteration v. s. energy difference in
the case of (11): constrained SSVQE. (f) number of iteration v.s. Energy
difference in the case of (12): constrained SSVQE with Tabu search.

##  IV Discussion

Our purpose is to obtain fine accuracy trace relationship between bonding
distance and energy on excited energy at the distance not to drop in local
minimum. The reason is because we need accurate physical and chemical constant
for chemical reaction.

VQE is developed for obtaining ground state energy. On the other hand, SSVQE
is developed for obtaining excited state energy. The error of ground state
energy is fewer than that of excited energy on VQE on H2. Moreover, the error
of excited energy is fewer than that of ground state on SSVQE on H2. The error
on HeH exhibits a similar tendency on VQE. However, the error on SSVQE on HeH
is different from VQE on H2 and it is very small. The ground and excited
states are both doublet on HeH. Hence, two degenerated states are derived at
once. Therefore, we think that it is important to fill degenarated orbitals or
create electron pair.

H2 molecules is stable as H atom consist of one proton and one electron and H2
molecules has 2s orbital that filled with two electrons. As the results,
analysis of energy state is enough to simulate on VQE. HeH is material
produced by nature and stable as positive ion. Excited energy system is
thought to be little bit unstable. To avoid the unstable states we need to
give restraint that is called constrained term. As the results, constrained
term is useful for analysis of ion. To obtain fine accuracy, we had better use
SSVQE method.

The constraint terms was beneficial for enhancing the accuracy as well as
decreasing errors. For the reasons mentioned above, Tabu search term was
useful by VQE methods on H2 and SSVQE method on HeH so as to decrease
convergence times. The reasson is because Tabu search is developed as
metaheurestic search algorithms, thus, Tabu search terms cut off the paths
destinated in local minimums in order to updates parameter sets pass the
shortest way to grobal minimum.

We are considering the charge balance on optimization depends on deviation of
whole molecules. We think that positive charge have an effect on the balance
of charge on HeH.

##  V Concluding Remarks

We investigated the effect of constraint and Tabu Search term on VQE and SSVQE
for ground and excited states. As a result, VQE method is adequate for the
calculation on H2 and SSVQE is adequate for the calculation on HeH,
respectively. Constraint and Tabu search terms contribute to enhance the
accuracy of energy levelsaccording to each bond length and to decrease
convergence times. According to the nature of molecules, we had better to use
VQE or SSVQE with constraint and Tabu search terms.Therefore, we suppose to
select adequate method (VQE or SSVQE) considering electron charge condition on
outer orbital.

Next subject is to improve the accuracies on calculation when objective
molecules have different orbital set. In order to achieve it, we had better
modify initial states and append extra terms though their trials are
formidable. Simplifying cluster and applying other optimizer that can search
the grobal minimum avoiding local them[26][27][28] are also. As a next step,
it is also worth to optimize energy levels of states by using Tabu search on
larger molecules. We will search suitable quantum algorithm such as VQE,
SSVQE, constraint and Tabu search term and so on when we solve energy state on
large molecules. We will select suitable method that depend on the nature of
molecules such as electron pair, balance of charge.

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