SlowGuess/NeurIPS_2023_OCR huggingface dataset:
which corresponds to the set on which the predicted uncertainty by GP is lower than a threshold  $\kappa$ . For sufficiently small  $\kappa$ , an appropriate kernel (which does not cause model misspecification), and a sufficiently weak prior, the GP predictions on CoRe are already accurate, and therefore CoRe can be regarded as "observed points." This leads to the following acquisition function, which we call Expected Maximum Improvement over Confident Regions (EMICoRe):
$$
a ^ {\mathrm {E M I C o R e}} \left(\boldsymbol {X} ^ {\prime}\right) = \frac {1}{M} \left\langle \max  \left(0, \min  _ {\boldsymbol {x} \in \mathcal {Z} _ {\boldsymbol {X}}} f (\boldsymbol {x}) - \min  _ {\boldsymbol {x} \in \mathcal {Z} _ {\bar {\boldsymbol {X}}}} f (\boldsymbol {x})\right) \right\rangle_ {p (f (\cdot) | \boldsymbol {X}, \boldsymbol {y})}, \tag {11}
$$
where  $\widetilde{\pmb{X}} = (\pmb{X},\pmb{X}^{\prime})\in \mathcal{X}^{N + M}$  denotes the augmented training set with the new input points  $X^{\prime}\in \mathcal{X}^{M}$ . This acquisition function evaluates the expected maximum improvement (per new observation point) when CoRe is expanded from  $\mathcal{Z}_{\pmb{X}}$  to  $\mathcal{Z}_{\widetilde{\pmb{X}}}$  by observing the objective at the new input points  $X^{\prime}$ . EMICoRe can be seen as a generalization of existing methods. If CoRe consists of the training points, it reduces to NEI [18]. If we set  $\kappa \rightarrow \infty$  so that the whole space is in the CoRe, and the random function  $f(\cdot)$  in Eq. (11) is replaced with its predictive mean of the current (first term) and the updated (second term) GP, EMICoRe reduces to knowledge gradient (KG) [30]. Thus, KG can be seen as a version of EMICoRe that ignores the uncertainty of the updated GP.
3.2.1 NFT-with-EMICoRe
We enhance the state-of-the-art NFT approach [11] with the VQE kernel and EMICoRe. We start from a brief overview of the NFT algorithm (detailed algorithms of NFT, NFT-with-EMICoRe, and the EMICoRe subroutine are given in Appendix F).
NFT: First, we initialize with a random point  $\hat{\pmb{x}}^0$  with  $\hat{y}^0 = f^* (\hat{\pmb{x}}^0) + \varepsilon$ . Then, for each iteration step  $t$ , we proceed as follows:
1. Select an axis  $d \in \{1, \dots, D\}$  sequentially or randomly and observe the objective  $\pmb{y}' = (y_1', y_2')^\top$  at deterministically chosen two points  $\pmb{X}' = (\pmb{x}_1', \pmb{x}_2') = \{\hat{\pmb{x}}^{t-1} - 2\pi/3\pmb{e}_d, \hat{\pmb{x}}^{t-1} + 2\pi/3\pmb{e}_d\}$  along the axis  $d$ .
2. Fit the sinusoidal function  $\widetilde{f}(\theta) = c_0 + c_1\cos\theta + c_2\sin\theta$  to the two new observations  $y_1', y_2'$  as well as the previous best estimated score  $\hat{y}^{t-1}$ . The optimal shift  $\hat{\theta}$  that minimizes  $\widetilde{f}(\theta)$  is analytically computed, which is used to get the new optimum  $\hat{\pmb{x}}^t = \hat{\pmb{x}}^{t-1} + \hat{\theta}\pmb{e}_d$ .
3. The best score is updated as  $\hat{y}^t = \tilde{f} (\hat{\boldsymbol{x}}^t)$
We stress that if the observation noise is negligible, i.e.,  $y \approx f(\pmb{x})$ , each step of NFT reaches the global optimum in the one-dimensional subspace along the chosen axis  $d$ , and thus performs SMO, see Proposition 2. In this case, the choice of the two new observation points  $\pmb{X}'$  is not important, as long as any pair of the three points are not exactly the same. However, when the observation noise is significant, the estimated global optimum in the chosen subspace is not necessary accurate, and the accuracy highly depends on the choice of the new observation points  $\pmb{X}'$ . In addition, errors can be accumulated in the best score  $\hat{y}^t$ , and therefore an additional measurement needs to be performed at  $\hat{\pmb{x}}^t$  after a certain iteration interval.
NFT-with-EMICoRe (ours): We propose to apply BO to NFT by using the VQE kernel and EMICoRe. Specifically, we use the GP regression model with the VQE kernel as a surrogate for BO, and choose new observation points  $\mathbf{X}^{\prime}$  by using the EMICoRe acquisition function. NFT-EMICoRe starts from  $T_{\mathrm{NFT}}$  NFT iterations until GP gets informative with a sufficient number of observations. After this initial phase, we proceed for each iteration  $t$  as follows:
1. Select an axis  $d \in \{1, \dots, D\}$  sequentially, and new observation points  $\mathbf{X}'$  by BO with EMICoRe, based on the previous optimum  $\hat{\pmb{x}}^{t-1}$ , the previous training data  $\{\pmb{X}^{t-1}, \pmb{y}^{t-1}\}$ , and the current CoRe threshold  $\kappa^t$  (this subroutine for EMICoRe will be explained below).
2. We observe  $\pmb{y}^{\prime}$  at the new points  $X^{\prime}$  chosen by EMICoRe, and train GP with the updated training data  $\pmb{X}^{t} = (\pmb{X}^{t - 1},\pmb{X}^{\prime}),\pmb{y}^{t} = (\pmb{y}^{t - 1^{\top}},\pmb{y}^{\prime^{\top}})^{\top}$ .
3. The subspace optimization is performed by fitting a sinusoidal function  $\widetilde{f}(\theta)$  to the GP posterior means  $\pmb{\mu} = (\mu(\hat{\pmb{x}}^{t-1} - 2\pi/3\pmb{e}_d), \mu(\hat{\pmb{x}}^{t-1}), \mu(\hat{\pmb{x}}^{t-1} + 2\pi/3\pmb{e}_d))^{\top}$  at three points. With the analytic minimum of the sinusoidal function,  $\hat{\theta} = \operatorname{argmin}_{\theta} \widetilde{f}(\theta)$ , the current optimum is computed:  $\hat{\pmb{x}}^t = \hat{\pmb{x}}^{t-1} + \hat{\theta}\pmb{e}_d$ .
4. For the current best score, we simply use the GP posterior mean  $\hat{\mu}^t = \mu (\hat{\pmb{x}}^t)$  and we set the new CoRe threshold to
$$
\kappa^ {t + 1} = \frac {\hat {\mu} ^ {t - T _ {\mathrm {A v e}}} - \hat {\mu} ^ {t}}{T _ {\mathrm {A v e}}}. \tag {12}
$$
Note that the GP posterior mean function lies in the VQE function space  $\mathcal{F}^{\mathrm{VQE}}$ , and therefore the fitting is done without error, and the choice of the three points in Step 3 does not affect the accuracy. Note also that the CoRe threshold  $\kappa^{t + 1}$  adjusts the required accuracy to the average reduction of the best score over the  $T_{\mathrm{Ave}}$  latest iterations—in the early phase where the energy  $\hat{\mu}^t$  decreases steeply, a large  $\kappa$  encourages crude optimization, while in the converging phase where the energy decreases slowly, a small  $\kappa$  enforces accurate optimization. Figure 1 illustrates the procedures of NFT and our EMICoRe approach.
The EMICoRe subroutine receives the previous optimum  $\hat{\pmb{x}}^{t - 1}$ , the previous training data  $\{X^{t - 1},y^{t - 1}\}$ , and the current CoRe threshold  $\kappa^t$ , and returns the new points  $X^{\prime}$  that maximizes EMICoRe. We fix the number of new samples to  $M = 2$ , and perform grid search along the chosen direction  $d$ . To this end,
1. We prepare  $J_{\mathrm{SG}}(J_{\mathrm{SG}} - 1)$  combinations of  $J_{\mathrm{SG}}$  search grid points  $\{\hat{\pmb{x}}^{t - 1} + \alpha_j\pmb {e}_d\}_{j = 1}^{J_{\mathrm{SG}}}$ , where  $\pmb {\alpha} = (\alpha_{1},\dots \alpha_{J_{\mathrm{SG}}})^{\top} = \frac{2\pi}{J_{\mathrm{SG}} + 1}\cdot (1,\dots,J_{\mathrm{SG}})^{\top}$ , as a candidate set  $\mathcal{C} = \{\check{\pmb{X}}^j\in \mathbb{R}^{D\times 2}\}_{j = 1}^{J_{\mathrm{SG}}(J_{\mathrm{SG}} - 1)}$ .
2. For each candidate  $\widetilde{\boldsymbol{X}}\in \mathcal{C}$ , we compute the updated GP posterior variance  $s_{\widetilde{\boldsymbol{X}}}(\boldsymbol {x},\boldsymbol {x}),\forall \boldsymbol {x}\in$ $\pmb{X}^{\mathrm{Grid}}$ , where  $s_{\widetilde{\boldsymbol{X}}}(\cdot ,\cdot)$  is the posterior covariance function of the GP trained on the augmented training points  $\widetilde{\pmb{X}} = (\pmb{X}^{t - 1},\check{\pmb{X}})$ , and  $\pmb{X}^{\mathrm{Grid}} = \{\hat{\pmb{x}}^{t - 1} + \alpha_j\pmb {e}_d\}_{j = 1}^{J_{\mathrm{OG}}}$  with  $\alpha = (\alpha_{1},\dots \alpha_{J_{\mathrm{OG}}})^{\top} = \frac{2\pi}{J_{\mathrm{OG}} + 1}\cdot (1,\dots,J_{\mathrm{OG}})^{\top}$  are  $J_{\mathrm{OG}}$  grid points along the axis  $d$ .
3. We obtain a discrete approximation to the updated CoRe as  $\mathcal{Z}_{\widetilde{\boldsymbol{X}}} = \{\boldsymbol{x} \in \boldsymbol{X}^{\mathrm{Grid}}; s_{\widetilde{\boldsymbol{X}}}(\boldsymbol{x}, \boldsymbol{x}) \leq \kappa^{2}\}$ . For simplicity, we approximate the previous CoRe to the previous optimum, i.e.,  $\mathcal{Z}_{\boldsymbol{X}^{t-1}} = \{\hat{\boldsymbol{x}}^{t-1}\}$ . After computing the mean and the covariance of the previous GP posterior,  $p(\hat{f}, \boldsymbol{f}^{\mathrm{test}} | \boldsymbol{X}^{t-1}, \boldsymbol{y}^{t-1})$ , at the previous best point  $\hat{\boldsymbol{x}}^{t-1}$  and the updated CoRe points (as the test set  $\boldsymbol{X}^{\mathrm{test}} = \mathcal{Z}_{\widetilde{\boldsymbol{X}}}$ )—which is  $\widetilde{D}$ -dimensional Gaussian for  $\widetilde{D} = |\mathcal{Z}_{\boldsymbol{X}^{t-1}} \cup \mathcal{Z}_{\widetilde{\boldsymbol{X}}}| = 1 + |\mathcal{Z}_{\widetilde{\boldsymbol{X}}}|$ —we estimate  $a_{\boldsymbol{X}^{t-1}}^{\mathrm{EMICoRe}} = \frac{1}{M} \langle \max \{0, \hat{f} - \min (\boldsymbol{f}^{\mathrm{test}})\} \rangle_{p(\hat{f}, \boldsymbol{f}^{\mathrm{test}} | \boldsymbol{X}^{t-1}, \boldsymbol{y}^{t-1})}$  by quasi Monte Carlo sampling.
The subroutine iterates this process for all candidates, and returns the best one,
$$
\boldsymbol{X}^{\prime} = \operatorname *{argmax}_{\check{\boldsymbol{X}}\in \mathcal{C}}a_{\boldsymbol{X}^{t - 1}}^{\mathrm{EMICoRe}}(\check{\boldsymbol{X}}).
$$
Parameter Setting Our approach has the crucial advantage that the sensitive parameters can be automatically tuned. The kernel smoothness parameter  $\gamma$  is optimized by maximizing the marginal likelihood of the GP, and the CoRe threshold  $\kappa$  is set to the average energy decrease of the last iterations as explained above. The noise variance  $\sigma^2$  is set by observing  $f^{*}(\pmb{x})$  several times at several random points and estimating  $\sigma^2 = \hat{\sigma}^{*2}(N_{\mathrm{shots}})$ . For the GP prior, the zero mean function  $\nu(\pmb{x}) = 0, \forall \pmb{x}$  is used, and the prior variance  $\sigma_0^2$  is roughly set so that the absolute value of the ground-state energy is in the same order as  $\sigma_0$ . Other parameters, including the number of grid points for search and CoRe discretization, should be set to sufficiently large values. See Appendix F for more details of parameter setting.
In Appendix F.4, we investigate other heuristics for the  $\kappa$  update, and find that the default update rule (12) performs comparably to the best heuristic.
3.3 Equivalence Between VQE Properties
Our method, NFT-with-EMICoRe, adapts BO to the VQE problem by harnessing one of the useful properties of VQE, namely the highly constrained objective function form (Proposition 2). One may now wonder if we can further improve NFT-EMICoRe by harnessing the other property, e.g., by regularizing GP so that its gradients follow the parameter shift rule (Proposition 1). We give a theory that answers to this question. Although the two properties were separately derived by analyzing the VQE objective (6), they are actually mathematically equivalent (the proof is given in Appendix G):
Theorem 2. For any periodic function  $f^{*}:[0,2\pi)^{D}\mapsto \mathbb{R}$ , Eq. (7)  $\Leftrightarrow$  Eq. (8).
This means that any sample from the GP prior with the VQE kernel (and any prior mean such that  $\nu(\cdot) \in \mathcal{F}^{\mathrm{VQE}}$ ) already satisfies the parameter shift rule. Theorem 2 implies that the parameter shift rule and the VQE function form are not two different properties, but two expressions of a single property, which is an important result that—to our knowledge—was not known in the literature.
We numerically demonstrate the performance of our approach for several setups, where the goal is to find the ground state of the quantum Heisenberg Hamiltonian
$$
H = - \left[ \sum_ {j = 1} ^ {Q - 1} \left(J _ {X} \sigma_ {j} ^ {X} \sigma_ {j + 1} ^ {X} + J _ {Y} \sigma_ {j} ^ {Y} \sigma_ {j + 1} ^ {Y} + J _ {Z} \sigma_ {j} ^ {Z} \sigma_ {j + 1} ^ {Z}\right) + \sum_ {j = 1} ^ {Q} \left(h _ {X} \sigma_ {j} ^ {X} + h _ {Y} \sigma_ {j} ^ {Y} + h _ {Z} \sigma_ {j} ^ {Z}\right) \right], \tag {13}
$$
where  $\{\sigma_{j}^{X},\sigma_{j}^{Y},\sigma_{j}^{Z}\}$  represent the Pauli matrices applied on the qubit in site  $j$ . The Heisenberg Hamiltonian, which represents a standard benchmarks of high practical relevance, is commonly used for evaluating the VQE performance (see, e.g., [38]),[VI] and its ground-truth ground state  $|\psi_{\mathrm{GS}}\rangle$  along with the ground-state energy  $\underline{f}^* = \langle \psi_{\mathrm{GS}}|H|\psi_{\mathrm{GS}}\rangle$  —which gives a lower-bound of the VQE objective (6)—can be analytically computed for small  $Q$ . For the variational quantum circuit  $G(\pmb {x})$  we use the  $L$ -layered Efficient SU(2) circuit (see Appendix C) with the open boundary, for which the search domain of the angular variables is  $\pmb {x}\in [0,2\pi)^{D}$  with  $D = (2 + (L\cdot 2))\cdot Q$ . We classically simulate the quantum computation with the Qiskit [43] library, and our Python implementation along with detailed tutorials on how to reproduce the results is publicly available on GitHub [44] at https://github.com/emicore/emicore.
To measure the optimization performance, i.e., the quality of the final solution  $\hat{\pmb{x}}^T$  after  $T$  iterations, we use two metrics: the true achieved lowest energy after  $T$  steps,  $\mathrm{ENG} \equiv f^{*}(\hat{\pmb{x}}^{T}) = \langle \psi_{0} | G(\hat{\pmb{x}}^{T})^{\dagger} H G(\hat{\pmb{x}}^{T}) | \psi_{0} \rangle$ , which is evaluated by simulating the noiseless observation with  $N_{\mathrm{shots}} \to \infty$ , and the fidelity FID  $\equiv \langle \psi_{\mathrm{GS}} | \psi_{\hat{\pmb{x}}^{T}} \rangle = \langle \psi_{\mathrm{GS}} | G(\hat{\pmb{x}}^{T}) | \psi_{0} \rangle$ , which measures how similar the best solution is to the true ground state. As the cost of observations, we count the total number of observed points, ignoring the cost of classical computation. We test each method 50 times, using the same set of initial points, for each method, for fair comparison. The initial points were randomly drawn from the uniform distribution in  $[0, 2\pi)^{D}$ , for fair comparisons. Details of the VQE circuits and the experimental settings can be found in Appendices C and H, respectively.
4.1 VQE-kernel Analysis
We first investigate the benefit of using the VQE-kernel for the following optimization problem: the task is to minimize the VQE objective (6) for the Ising Hamiltonian, a special case of the Heisenberg Hamiltonian with the coupling parameters set to  $J_{X} = -1$ ,  $J_{Y} = J_{Z} = 0$ ,  $h_{X} = h_{Y} = 0$ ,  $h_{Z} = -1$ , with the  $(L = 3)$ -layered quantum circuit with  $(Q = 3)$ -qubits. Namely, we solve the problem (1) in the  $(D = 2(L + 1)Q = 24)$ -dimensional space. We use the standard BO procedure with GP regression, and compare our VQE-kernel with the Gaussian radial basis function (RBF) [32] and the periodic kernel [17], where the EI acquisition function is maximized by L-BFGS [45]. Figure 2 shows the achieved energy (left) and the fidelity (right) with different kernels. In each panel, the left
$^{\mathrm{VI}}$ Spin chain Hamiltonians are widely studied in condensed matter physics [39], and generalized spin chains also emerge from the lattice discretization of field theories in low dimensions (see, e.g., Ref. [40] or Refs. [41, 42] for reviews).
Figure 2: Comparison of our VQE-kernel (red) to the RBF and the periodic kernel benchmarks (blue and orange) in the VQE optimization using the standard BO procedure, for the Ising Hamiltonian with the  $(L = 3)$ -layered  $(Q = 3)$ -qubits quantum circuit. The search domain dimension is  $D = 24$ , and  $N_{\mathrm{shots}} = 1024$  readout shots are taken for each observation. The energy (left) and the fidelity (right) are plotted, and in each plot, optimization progress is shown with the median (solid) and the 25- and 75-th percentiles (shadows) over 50 trials. The portrait square shows the distribution of the final solution after 150 observations have been performed.
Figure 3: Comparison (in the same format as Figure 2) between our EMICoRe (red) and the NFT baselines (green and purple) in the VQE for the Ising (top row) and Heisenberg (bottom row) Hamiltonians with the  $(L = 3)$ -layered  $(Q = 5)$ -qubits quantum circuit (thus,  $D = 40$ ) and  $N_{\mathrm{shots}} = 1024$ . We confirmed for the Ising Hamiltonian that longer optimization by EMICoRe up to 6000 observed points reaches the ground state with  $98\%$  fidelity (see Appendix I.1).
plot shows the progress of optimization (median as a solid curve and the 25- and 75-th percentiles as shadows) as a function of the observation cost, i.e., the number of observed points after corresponding iterations. The portrait square on the right shows the distribution (by kernel density estimation [46]) of the best solutions after 150 observations have been performed. We see that the proposed VQE kernel (red), which achieves  $0.93 \pm 0.05$  fidelity, converges more stably than the baseline RBF and periodic kernels, both of which achieve  $0.90 \pm 0.09$  fidelity. Therefore, the VQE kernel leads to better statistical efficiency albeit its effect appears rather mild in the regime of a small number of qubits.
4.2 NFT-with-EMICoRe Analysis
We now evaluate our NFT-with-EMICoRe approach and show that this improves the state-of-the-art baselines of NFT [11]. Specifically, we compare our method with two versions of NFT: NFT-Sequential updates along each axis sequentially, while NFT-Random chooses the axis randomly. Figure 3 shows the results of VQE for the Ising Hamiltonian (top row) and the Heisenberg Hamiltonian (bottom row) with the parameters set to  $J_{X} = J_{Y} = J_{Z} = 1$  and  $h_{X} = h_{Y} = h_{Z} = 1$ . For both
Figure 4: Evolution of the GP in NFT-with-EMICoRe. The posterior mean (blue solid) with uncertainty (blue shadow) along the direction  $d$  to be optimized in the steps  $t = 0,1,293,294$  (columns) are shown before (top row) and after (bottom row) the chosen two new points  $X' = (x_1', x_2')$  are observed. The red solid curve shows the true energy  $f^*(\pmb{x})$ . On the top, the number of observed samples  $|\pmb{X}|$  until step  $t - 1$  and the direction  $d$  are shown.
Hamiltonians, we use  $(L = 3)$ -layered  $(Q = 5)$ -qubits variational quantum circuits, thus  $D = 40$  with each observation performed by  $N_{\mathrm{shots}} = 1024$  readout shots. The figure shows that our NFT-with-EMICoRe method consistently outperforms the baselines in terms of both achieved energy (left) and fidelity (right). Moreover, the variance over the trials is reduced, implying its robustness against the initialization. We conducted additional experiments to answer the important questions of whether our approach converges to the ground state with high fidelity, and how the Hamiltonian, the number of qubits, and the number of readout shots affect the optimization performance. The results reported in Appendix I show that our NFT-with-EMICoRe method reaches the ground state with  $98\%$  fidelity after 6000 observed points, see Appendix I.1, and consistently outperforms the baseline methods in various settings of Hamiltonians, qubits, and the number of readout shots, see Appendices I.2 and I.3. Thus, we conclude that our approach of combining NFT and BO with the VQE-kernel and the EMICoRe acquisition function is suited for efficient VQE optimization, dealing with different levels of observation noise.
Figure 4 shows the evolution of the GP during optimization by NFT-with-EMICoRe. The blue curve and the shadow show the posterior mean (solid) and the uncertainty (shadow) at four different steps  $t$  (columns). The top and the bottom rows show the GP before and after the two new points  $\pmb{X}^{\prime} = (\pmb{x}_{1}^{\prime},\pmb{x}_{2}^{\prime})$ , chosen by EMICoRe, are observed. The red solid curve shows the true energy  $f^{*}(\pmb{x})$ . We observe the following: In the early phase (the left two columns), GP is uncertain except at the current optimum  $\hat{\pmb{x}}^t$  before new points are observed; After observing the chosen two points  $\pmb{X}^{\prime}$ , GP is certain on the whole subspace, thanks to the VQE kernel; and in the converging phase (the right two columns), GP is certain over the whole subspace even before new observations, and the fine-tuning of the angular variable  $\pmb{x}$  is performed.
Efficient, noise-resilient algorithms for optimizing hybrid quantum-classical algorithms are pivotal for the NISQ era. In this paper, we propose EMICoRe, a new Bayesian optimization approach, in combination with a novel VQE-kernel which leverages strong inductive biases from physics. Our physics-informed search with EMICoRe achieves faster and more stable convergence to the minimum energy, thus identifying optimal quantum circuit parameters. Remarkably, the physical inductive bias makes the optimization more resilient to observation noise. The insight that more physical information helps to gain a better practical understanding of quantum systems is a starting point for further algorithmic improvements and further exploration of other possible physical biases. This may ultimately lead to more noise-resilient algorithms for quantum circuits, thus enabling the accessibility of larger-scale experiments in the quantum computing realm. In future work, we plan to investigate the effect of hardware-dependent noise, and test our approach on real quantum devices.
The authors thank the referees for their insightful comments, which significantly improved the paper. This work was partially supported by the German Ministry for Education and Research (BMBF) as BIFOLD - Berlin Institute for the Foundations of Learning and Data (BIFOLD23B), the Einstein Research Unit (ERU) Quantum Project (ERU-2020-607), and the European Union's HORIZON MSCA Doctoral Networks programme and the AQTIVATE project (101072344). This work is supported with funds from the Ministry of Science, Research and Culture of the State of Brandenburg within the Centre for Quantum Technologies and Applications (CQTA). PS acknowledges support from ERC AdG NOQIA; MICIN/AEI (PGC2018-0910.13039/501100011033, CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, FPI). K.-R.M. was partly supported by the Institute of Information & Communications Technology Planning & Evaluation (IITP) grants funded by the government (MSIT) (No. 2019-0-00079, Artificial Intelligence Graduate School Program, Korea University and No. 2022-0- 00984, Development of Artificial Intelligence Technology for Personalized Plug-and-Play Explanation and Verification of Explanation).