SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figure 5: Illustration of the VQE workflow. In the first step, highly complicated optimization problems can be translated into the Hamiltonian formulation (see, e.g., [50, 51]). The Hamiltonian  $H$  and an initial state  $|\psi_0\rangle$  are plugged into the VQE block (light purple), where the variational quantum circuit is instantiated with random angular parameters  $\mathbf{x}^0$ . In the VQE block, the top vertex of the triangle represents the quantum computer, the red arrows and blocks refer to operations running on a quantum computer, while the green parts refer to classical steps. In the bottom-left green box, the current parameters  $\mathbf{x}$  are updated with the new best parameters  $\hat{\mathbf{x}}$  found during classical optimization routines. Then, the quantum circuit  $G(\mathbf{x})$  is updated using the new optimum point,  $\mathbf{x} \rightarrow \hat{\mathbf{x}}$ , and the energy  $E(\mathbf{x})$  for the updated variational wave function  $|\psi_{\mathbf{x}}\rangle$  is measured. The VQE block is executed for  $T$  iterations and finally outputs the solution to the task as a variational approximation  $|\psi_{\hat{\mathbf{x}}^T}\rangle$  of the ground state and the corresponding optimal parameters  $\hat{\mathbf{x}}^T$  for the quantum circuit.
Our EMICoRe, proposed in Section 3.2, can be seen as a generalization of NEI. In EMICoRe, the points in the confident regions (CoRe), where the predictive uncertainty after new points would have been observed is lower than a threshold  $\kappa$ , are treated as "indirectly observed", and the best score is searched for over CoRe. If we replace CoRe with the previous and the new training points, EMICoRe reduces to NEI.
Another related method to our approach is Knowledge Gradient (KG) [30]:
$$
\left. a _ {\boldsymbol {X}} ^ {\mathrm {K G}} \left(\boldsymbol {X} ^ {\prime}\right) = \left\langle \max  \left(0, \min  _ {\boldsymbol {x} ^ {\prime \prime} \in \mathcal {X}} \mu_ {\boldsymbol {X}} \left(\boldsymbol {x} ^ {\prime \prime}\right) - \min  _ {\boldsymbol {x} ^ {\prime \prime} \in \mathcal {X}} \mu_ {\left(\boldsymbol {X}, \boldsymbol {X} ^ {\prime}\right)} \left(\boldsymbol {x} ^ {\prime \prime}\right)\right) \right\rangle_ {p \left(\boldsymbol {y} ^ {\prime} \mid \boldsymbol {X}, \boldsymbol {y}\right)}, \right. \tag {25}
$$
which assumes that the minimizer of the GP posterior mean function is used as the best score—even if the uncertainty at the minimizer is high—and estimates the improvement of the best score before and after the new points  $\mathbf{X}'$  are added to the training data. The second term in Eq. (25) is estimated by simulation: it trains the GP on the augmented data  $(\mathbf{X}, \mathbf{X}')$  and  $(\mathbf{y}^\top, \mathbf{y}'^\top)^\top$ , where  $\mathbf{y}'$  are drawn from the GP posterior  $p(\mathbf{y}'|\mathbf{X}, \mathbf{y})$ , and finds the minimizer of the updated GP mean. Iterating this process provides Monte Carlo samples to estimate the second term in Eq. (25).
In our EMICoRe method, if we set the CoRe threshold  $\kappa^2\rightarrow \infty$  so that the entire search domain is in CoRe, and replace the random function  $f(\cdot)$  in Eq. (11) with its previous and updated GP means, respectively, EMICoRe reduces to KG. Thus, KG can be seen as a version of EMICoRe that ignores the uncertainty of the updated GP.
C Details of Variational Quantum Eigensolvers (VQEs)
The VQE [7, 8] is a hybrid quantum-classical algorithm that uses a classical optimization routine in conjunction with a quantum processor to approximate the ground state of a given Hamiltonian. VQEs are designed to run on Noisy Intermediate-Scale Quantum (NISQ) devices, which are the current generation of quantum computers. These devices have a limited number of qubits and high error rates, which makes it challenging to execute complex quantum algorithms faithfully on these quantum devices. VQEs are a promising approach to tackle this challenge since they use a hybrid classical-quantum algorithm, where the quantum device only needs to perform relatively simple computations (see Fig. 5 for an illustration of the VQE workflow).
VQEs are considered to be potentially relevant for some challenging problems in different scientific domains such as quantum chemistry [52, 53, 54], drug discovery [55, 56, 57], condensed matter physics [58], materials science [59] and quantum field theories [60, 61, 62, 41]. Specifically, finding
the ground state of a molecule is a very challenging problem growing exponentially in complexity with the number of atoms for classical computing, while for VQE this would instead scale polynomially. Despite being naturally designed to solve problems associated with quantum chemistry and physics, such as calculating molecular energies, optimizing molecular geometries [63], and simulating strongly correlated systems [64], VQEs have also been applied to other domains including optimization and combinatorial problems such as the flight gate assignment [50, 51].
Given a Hamiltonian formulation that can be efficiently measured on a quantum device, the variational approach of VQE can be applied to obtain an upper bound for the ground-state energy as well as an approximation for the ground-state wave function. Let  $|\psi_0\rangle$  be the initial ansatz for the  $Q$ -dimensional (qubit) wave function. Let us assume that we use a parametrized quantum circuit  $G(\pmb{x})$ , where  $\pmb{x} \in [0, 2\pi)^D$  represents the angular parameters of quantum gates. The circuit  $G$  consists of  $D'(\geq D)$  unitary gates:
$$
G (\boldsymbol {x}) = G _ {D ^ {\prime}} \circ \dots \circ G _ {1}, \tag {26}
$$
where  $D$  of the  $D^{\prime}$  gates depend on one of the angular parameters exclusively, i.e.,  $x_{d}$  parametrizes only a single gate  $G_{d^{\prime}(d)}(x_d)$ , where  $d^{\prime}(d)$  specifies the gate parametrized by  $x_{d}$ . We consider the parametric gates of the form
$$
G _ {d ^ {\prime}} (x) = U _ {d ^ {\prime}} (x) = \exp \left\{- i \frac {x}{2} P _ {d ^ {\prime}} \right\}, \tag {27}
$$
where  $P_{d'}$  is an arbitrary sequence of the Pauli operators  $\{\sigma_q^X, \sigma_q^Y, \sigma_q^Z\}_{q=1}^Q$  acting on each qubit at most once. This form covers not only single-qubit gates such as  $R_X(x) = \exp(-i\theta \sigma_q^X)$ , but also entangling gates such as  $R_{XX}(x) = \exp(-ix\sigma_{q_1}^X \circ \sigma_{q_2}^X)$  and  $R_{ZZ}(x) = \exp(-ix\sigma_{q_1}^Z \circ \sigma_{q_2}^Z)$  for  $q_1 \neq q_2$ . In the matrix representation of quantum mechanics, quantum states are expressed as vectors in the computational basis, i.e.,
$$
\langle 0 | = (1 \quad 0), \quad | 0 \rangle = \binom {1} {0}, \quad \langle 1 | = (0 \quad 1), \quad | 1 \rangle = \binom {0} {1}. \tag {28}
$$
Moreover, Pauli operators are expressed as matrices,
$$
\sigma^ {X} = \left( \begin{array}{c c} 0 & 1 \\ 1 & 0 \end{array} \right), \qquad \qquad \sigma^ {Y} = \left( \begin{array}{c c} 0 & - i \\ i & 0 \end{array} \right), \qquad \qquad \sigma^ {Z} = \left( \begin{array}{c c} 1 & 0 \\ 0 & - 1 \end{array} \right),
$$
acting only on one of the  $Q$  qubits non-trivially. The application of any operator on a quantum state  $|\psi \rangle$  thus becomes a matrix multiplication in the chosen basis. Single-qubit parametric gates correspond to rotations around the axes in a three-dimensional space representation of a qubit state, known as the Bloch sphere, and are widely used in ansatz circuits for VQEs.
Given the Hamiltonian  $H$ , which is an Hermitian operator, the quantum device measures the energy of the resulting quantum state  $|\psi_{\pmb{x}}\rangle = G(\pmb{x})|\psi_0\rangle$  contaminated with observation noise  $\varepsilon$ , i.e.,
$$
f (\boldsymbol {x}) = f ^ {*} (\boldsymbol {x}) + \varepsilon , \quad \text {w h e r e} \quad f ^ {*} (\boldsymbol {x}) = \left\langle \psi_ {\boldsymbol {x}} | H | \psi_ {\boldsymbol {x}} \right\rangle = \left\langle \psi_ {0} | G (\boldsymbol {x}) ^ {\dagger} H G (\boldsymbol {x}) | \psi_ {0} \right\rangle . \tag {29}
$$
The observation noise  $\varepsilon$  in our numerical experiments only incorporates the shot noise coming from the intrinsically probabilistic nature of quantum measurements, and the errors from imperfect qubits, gates, and measurements on current NISQ devices are not considered.
The task of the classical computer in VQE is to find the optimal angular parameters  $\pmb{x}$  such that  $f^{*}(\pmb{x})$  is minimal. Given that the ansatz circuit is expressive enough,  $G(\pmb{x})|\psi_0\rangle$  then corresponds to the ground state of the Hamiltonian  $H$ . Thus, the problem to be solved is a noisy black-box optimization:
$$
\min  _ {\boldsymbol {x} \in [ 0, 2 \pi) ^ {D}} f ^ {*} (\boldsymbol {x}).
$$
The long-term goal of research on VQEs is to develop efficient quantum algorithms that can solve problems beyond the capabilities of classical computers. While VQE is a promising approach for exploiting the advantages of quantum computing, they are plagued by some limitations. Specifically, these algorithms require appropriate choices for the quantum circuit [65, 66]. To have compact circuits with a moderate number of quantum gates is, therefore, essential to favor stable computations and high measurement accuracy.
$^{\mathrm{VI}}$ In Appendix E, we discuss how the theorems and our method can be extended to the non-exclusive parametrization case.
Figure 6: Illustration of Qiskit's [43] Efficient SU(2) Circuit with default parameters for  $Q = 3$  qubits and  $L = 2$  layers (thus  $\pmb{x} \in [0,2\pi)^{18}$ ). Each dashed box indicates a layer of the ansatz circuit. The quantum computation proceeds from left to right with an initial ansatz of the form  $|0\rangle^{\otimes Q}$ . Each horizontal line corresponds to a quantum wire representing one qubit. The symbols on the wires correspond to gate operations on the respective qubits, starting with two parametrized rotational gates  $R_{Y}$  and  $R_{Z}$  acting on the qubits in the initial 0-th layer. Then, each of the following layers is composed of one block of CNOT gates and two rotational gates acting on each qubit.
Efficient SU(2) circuit: One circuit ansatz commonly used is the Efficient SU(2) Circuit from Qiskit [43], which is illustrated in Figure 6. In this circuit, two consecutive rotational gates,
$$
R _ {Y} (x) = \exp \left\{- i \frac {x}{2} \sigma^ {Y} \right\} \quad \text {a n d} \quad R _ {Z} (x) = \exp \left\{- i \frac {x}{2} \sigma^ {Z} \right\}, \tag {30}
$$
act on each qubit in the initial (0-th) layer. The rest of the circuit is composed by a stack of  $L$  layers, each of which consists of CNOT gates applied to all pairs of qubits, and a pair of the rotational gates,  $R_{Y}$  and  $R_{Z}$ , acting on each qubit. The CNOT gates are not parametrized and are therefore not updated during the optimization process. Therefore, the total number of angular parameters of the circuit is equal to the number of  $R_{Y}$  and  $R_{Z}$  gates in the circuit. In total, for a setup having  $Q$  qubits and  $L$  layers, the number of angular parameters is
$$
D = 2 \times Q + (L \times 2) \times Q, \tag {31}
$$
where the first term counts the rotational gates in the initial layer, while the second term counts those in the latter layers. In our experiments, we set the initial ansatz to  $|\psi_0\rangle = |0\rangle^{\otimes Q}$ , which corresponds to the tensor product of  $Q$  qubits in the  $|0\rangle$ -ket state. $^{VIII}$
Proof. The VQE kernel (9) can be rewritten as
$$
\begin{array}{l} k ^ {\mathrm {V Q E}} (\pmb {x}, \pmb {x} ^ {\prime}) = \sigma_ {0} ^ {2} (\gamma^ {2} + 2) ^ {- D} \prod_ {d = 1} ^ {D} \left(\gamma^ {2} + 2 \cos (x _ {d} - x _ {d} ^ {\prime})\right) \\ = \sigma_ {0} ^ {2} (\gamma^ {2} + 2) ^ {- D} \prod_ {d = 1} ^ {D} \left(\gamma^ {2} + 2 \cos x _ {d} \cos x _ {d} ^ {\prime} + 2 \sin x _ {d} \sin x _ {d} ^ {\prime}\right) \\ = \sigma_ {0} ^ {2} (\gamma^ {2} + 2) ^ {- D} \sum_ {\xi \in \{0, 1, 2 \} ^ {D}} \prod_ {d = 1} ^ {D} (\gamma^ {2}) ^ {\mathbb {1} (\xi_ {d} = 0)} (2 \cos x _ {d} \cos x _ {d} ^ {\prime}) ^ {\mathbb {1} (\xi_ {d} = 1)} (2 \sin x _ {d} \sin x _ {d} ^ {\prime}) ^ {\mathbb {1} (\xi_ {d} = 2)} \\ = \phi (\boldsymbol {x}) ^ {\top} \phi (\boldsymbol {x} ^ {\prime}), \\ \end{array}
$$
where  $\mathbb{1}(\cdot)$  denotes the indicator function (equal to one if the event is true and zero otherwise), and
$$
\boldsymbol {\phi} (\boldsymbol {x}) = \sigma_ {0} (\gamma^ {2} + 2) ^ {- D / 2} \cdot \mathbf {v e c} \left(\otimes_ {d = 1} ^ {D} (\gamma , \sqrt {2} \cos x _ {d}, \sqrt {2} \sin x _ {d}) ^ {\top}\right) \in \mathbb {R} ^ {3 ^ {D}},
$$
E Generalization to Non-Exclusive Parametrization Case
In the VQE, it is often beneficial to share the same parameter amongst multiple gates, for example, in the case where the Hamiltonian has a certain symmetry, such as translation invariance. Doing so,
An equivalent notation often found in the literature is  $|0\rangle^{\otimes Q} = |\mathbf{0}\rangle$
the variational quantum circuit is guaranteed to generate quantum states that fulfill the symmetry, and thus the number of parameters to be optimized is efficiently reduced. To consider this case, we need to assume that some of the entries of the search variable  $\pmb{x}$  are shared parameters, each of which parametrizes multiple gates. The Nakanishi-Fujii-Todo (NFT) algorithm [11] can still be used in this case, based on the following generalization of Proposition 2:
Proposition 3. [11] Assume that the  $d$ -th entry of the input  $\pmb{x} \in [0,2\pi)^{D}$  parametrizes  $V_{d} \geq 1$  gate parameters. Then, for the VQE objective function  $f^{*}(\cdot)$  in Eq. (6),
$$
\exists \boldsymbol {b} \in \mathbb {R} ^ {\prod_ {d = 1} ^ {D} (1 + 2 V _ {d})} \quad \text {s u c h t h a t} \quad f ^ {*} (\boldsymbol {x}) = \boldsymbol {b} ^ {\top} \cdot \mathbf {v e c} \left(\otimes_ {d = 1} ^ {D} \left( \begin{array}{c} 1 \\ \cos x _ {d} \\ \vdots \\ \cos \left(V _ {d} x _ {d}\right) \\ \sin x _ {d} \\ \vdots \\ \sin \left(V _ {d} x _ {d}\right) \end{array} \right)\right). \tag {32}
$$
Similarly, our VQE kernel introduced in Theorem 1 can be generalized as follows:
Theorem 3. The (higher-order) VQE kernel,
$$
k ^ {V Q E} (\boldsymbol {x}, \boldsymbol {x} ^ {\prime}) = \sigma_ {0} ^ {2} \prod_ {d = 1} ^ {D} \left(\frac {\gamma^ {2} + 2 \sum_ {v = 1} ^ {V _ {d}} \cos (v \left(x _ {d} - x _ {d} ^ {\prime}\right))}{\gamma^ {2} + 2 V _ {d}}\right), \tag {33}
$$
is decomposed as  $k^{VQE}(\pmb{x},\pmb{x}') = \phi (\pmb{x})^\top \phi (\pmb{x}')$ , where
$$
\phi (\boldsymbol {x}) = \sigma_ {0} \left(\gamma^ {2} + 2 V _ {d}\right) ^ {- D / 2} \cdot \mathbf {v e c} \left(\otimes_ {d = 1} ^ {D} \left( \begin{array}{c} \gamma \\ \sqrt {2} \cos x _ {d} \\ \vdots \\ \sqrt {2} \cos \left(V _ {d} x _ {d}\right) \\ \sqrt {2} \sin x _ {d} \\ \vdots \\ \sqrt {2} \sin \left(V _ {d} x _ {d}\right) \end{array} \right)\right). \tag {34}
$$
Proof. Similarly to the exclusive parameterization (or first-order) case, we have
$$
\begin{array}{l} k ^ {\mathrm {V Q E}} (\pmb {x}, \pmb {x} ^ {\prime}) = \sigma_ {0} ^ {2} \prod_ {d = 1} ^ {D} \left(\frac {\gamma^ {2} + 2 \sum_ {v = 1} ^ {V _ {d}} \cos (v (x _ {d} - x _ {d} ^ {\prime}))}{\gamma^ {2} + 2 V _ {d}}\right) \\ = \sigma_ {0} ^ {2} \left(\gamma^ {2} + 2 V _ {d}\right) ^ {- D} \prod_ {d = 1} ^ {D} \left(\gamma^ {2} + 2 \sum_ {v = 1} ^ {V _ {d}} \left\{\cos \left(v x _ {d}\right) \cos \left(v x _ {d} ^ {\prime}\right) + \sin \left(v x _ {d}\right) \sin \left(v x _ {d} ^ {\prime}\right) \right\}\right) \\ = \sigma_ {0} ^ {2} \left(\gamma^ {2} + 2 V _ {d}\right) ^ {- D} \\ \end{array}
$$
$$
\begin{array}{l} \sum_ {\boldsymbol {\xi} \in \{0, \dots , 2 V _ {d} \} ^ {D}} \prod_ {d = 1} ^ {D} (\gamma^ {2}) ^ {\mathbb {1} (\xi_ {d} = 0)} \prod_ {v = 1} ^ {V _ {d}} \left(2 \cos (v x _ {d}) \cos (v x _ {d} ^ {\prime})\right) ^ {\mathbb {1} (\xi_ {d} = 2 v - 1)} \left(2 \sin (v x _ {d}) \sin (v x _ {d} ^ {\prime})\right) ^ {\mathbb {1} (\xi_ {d} = 2 v)} \\ = \phi (\boldsymbol {x}) ^ {\top} \phi (\boldsymbol {x} ^ {\prime}), \\ \end{array}
$$
Since the VQE objective (32) is the  $V_{d}$ -th order sinusoidal function along the  $x_{d}$ -axis, NFT can perform the sequential minimal optimization (SMO) by observing  $2V_{d}$  points in each step — which become  $2V_{d} + 1$  points, together with the current optimum point — to determine the entire function form in the one-dimensional subspace parallel to the  $x_{d}$ -axis. Our NFT-with-EMICoRe approach can similarly be generalized to the non-exclusive cases: for the chosen direction  $d$ , the approach observes  $M = 2V_{d}$  new points by maximizing the EMICoRe acquisition function, based on the GP regression with the generalized VQE kernel (33).
Algorithm 1: Nakanishi-Fuji-Todo (NFT) method [11] (Baseline)
input:  $T_{\mathrm{MI}}$  : max # of iterations  $T_{\mathrm{RI}}$  : reset interval  $\mathcal{D}^0 = (\hat{x}^0,\hat{y}^0)$  : initialization with  $\hat{\pmb{x}}^0\sim [0,2\pi)^D$  and  $\hat{y}^0 = f^* (\hat{x}^0) + \varepsilon$    
output:  $\hat{\pmb{x}}^{T_{\mathrm{MI}}}$  : last optimal point  $y(\hat{x}^{T_{\mathrm{MI}}})$  : last estimated objective  $\mathcal{D}^{T_{\mathrm{MI}}}$  : ensemble of collected observations   
1 begin   
for  $t = 1$  to  $T_{\mathrm{MI}}$  do Choose a direction  $d\in \{1,\dots ,D\}$  sequentially or randomly; Find  $(X^{\prime})\Rightarrow X^{\prime} = (x_1^{\prime},x_2^{\prime}) = \{\hat{x}^{t - 1} - 2\pi /3e_d,\hat{x}^{t - 1} + 2\pi /3e_d\}$  along d; Observe  $\pmb {y}'\Longrightarrow \pmb {y}' = (y_1',y_2')^\top$  at the new points  $X^{\prime}$  . Append  $(\mathcal{D}^{t - 1}\cup (\mathbf{X}',\mathbf{y}')\Rightarrow \mathcal{D}^t$  Fit  $(\widetilde{f} (\theta))\Rightarrow \widetilde{f} (\theta) = c_0 + c_1\cos \theta +c_2\sin \theta$  to the three points  $\{(-2\pi /3,y_1'),(0,\hat{y}^{t - 1}),(2\pi /3,y_2')\}$  ; FindMin  $(\widetilde{f})\Longrightarrow$  find analytical minimum  $\hat{\theta} = \mathrm{argmin}_{\theta \in [0,2\pi ]}\widetilde{f} (\theta)$  Update  $\hat{\pmb{x}}\Longrightarrow$  with  $\hat{\pmb{x}}^t = \hat{\pmb{x}}^{t - 1} + \hat{\pmb{\theta}} e_d;$  Update  $(\hat{y})\Longrightarrow$  with estimated  $\hat{y}^t = \tilde{f} (\hat{\theta})$  if t mod  $T_{\mathrm{RI}} = 0$  then Observe  $y(\hat{x}^t)$  /\* Perform additional observation \*/  $D^{t} = D^{t}\cup (\hat{x}^{t},y(\hat{x}^{t}))$  end   
end   
17 return  $D^{T_{\mathrm{MI}}},\hat{x}^{T_{\mathrm{MI}}},y(\hat{x}^{T_{\mathrm{MI}}})$
F Algorithm Details
Here, we provide the detailed procedures of the baseline method (Nakanishi-Fuji-Todo (NFT) [11]) and our proposed method (NFT-with-EMICoRe and the EMICoRe subroutine).
F.1 Nakanishi-Fuji-Todo (NFT)
In each step of NFT (Algorithm 1), an axis  $d \in \{1, \dots, D\}$  is chosen sequentially or randomly, and the next observation points along the axis are set (Step 4) and observed (Step 5). Based on Proposition 2, the two new observations together with the previous optimum are fitted to a sinusoidal function, and the global optimum along the axis is analytically computed, establishing sequential minimal optimization (SMO) [12]. NFT iterates this process from a random initial point and outputs the collected datapoints and the last optimum. Since the optimal point at each step is not directly observed, errors can accumulate over iterations. As a remedy, NFT observes the optimal point when the reset interval condition is met (Step 12).
F.2 NFT-with-EMICoRe
Our proposed method, NFT-with-EMICoRe (Algorithm 2), replaces the deterministic choice of the next observation points  $\mathbf{X}'$  in NFT with a promising choice by BO. After initial (optional) iterations of NFT until sufficient training data are collected, we start the EMICoRe subroutine (Algorithm 3) to suggest new observation points  $\mathbf{X}'$  by BO (Step 8). Then, the objective values  $\mathbf{y}'$  at  $\mathbf{X}'$  are observed (Step 10), and the training data  $\mathcal{D}^{t - 1} = \{\mathbf{X}^{t - 1},\mathbf{y}^{t - 1}\}$  are updated with the new observed data
Algorithm 2: NFT-with-EMICoRe
input:  $T_{\mathrm{MI}}:\# \mathrm{of}$  iterations  $T_{\mathrm{NFT}}:\#$  of initial NFT steps  $T_{\mathrm{Ave}}(>T_{\mathrm{NFT}}):$  averaging steps for  $\kappa$  update.  $\mathcal{D}^0 = (\hat{\pmb{x}}^0,\hat{\pmb{y}}^0)$  : initialization with  $\hat{\pmb{x}}^0\sim [0,2\pi)^D$  and  $\hat{y}^0 = f^* (\hat{x}^0) + \varepsilon$    
output:  $\hat{\pmb{x}}^{T_{\mathrm{MI}}}$  : last optimal point  $\mu (\hat{\pmb{x}}^{T_{\mathrm{MI}}})$  : last estimated objective  $\mathcal{D}^{T_{\mathrm{MI}}}$  : ensemble of collected observations   
if  $T_{\mathrm{NFT}} > 0$  then  $\mathcal{D}^{T_{\mathrm{NFT}}}_,\_ ,\_ = \mathrm{NFT}(T_{\mathrm{MI}} = T_{\mathrm{NFT}},T_{\mathrm{RI}} = 0,D);$  /\* see Algorithm 1 \*/ Update observations  $\mathcal{D}^{T_{\mathrm{NFT}}} = \mathcal{D}\cup \mathcal{D}^0$  /\* Collect points from NFT step \*/   
end   
begin   
for  $t = T_{\mathrm{NFT}} + 1$  to  $T_{\mathrm{MI}}$  do Choose a direction  $d\in \{1,\dots ,D\}$  sequentially or randomly; Find  $X^{\prime}\Rightarrow$  points maximizing the EMICoRe  $(\hat{\pmb{x}}^{t - 1},\mathcal{D}^{t - 1},d^{t},\kappa^{t})$  . /for EMICoRe sub-routine see Algorithm 3 \*/ Observe  $\pmb {y}'\Rightarrow \pmb {y}' = (y_1',y_2')^\top$  at the new points  $X^{\prime}$  . Append  $(\mathcal{D}^{t - 1}\cup (\pmb {X}',\pmb {y}'))\Rightarrow \mathcal{D}^t$  Train GP  $(\mathcal{D}^t)$  on updated dataset; Compute posterior means  $\mu = (\mu (\hat{\pmb{x}}^{t - 1} - 2\pi /3e_d),\mu (\hat{\pmb{x}}^{t - 1}),\mu (\hat{\pmb{x}}^{t - 1} + 2\pi /3e_d))^{\top};$  Fit  $(\widetilde{f} (\theta))\Rightarrow \widetilde{f} (\theta) = c_0 + c_1\cos \theta +c_2\sin \theta$  to the three points  $\left\{(-2\pi /3,\mu_1),(0,\mu_2),(2\pi /3,\mu_3)\right\}$  . FindMin  $(\widetilde{f})\Rightarrow$  find analytical min  $\hat{\theta} = \operatorname *{argmin}_{\theta \in [0,2\pi ]}\widetilde{f} (\theta)$  . Update  $\hat{\pmb{x}}\Rightarrow$  with  $\hat{\pmb{x}}^t = \hat{\pmb{x}}^{t - 1} + \hat{\theta}\pmb {e}_d$  Evaluate optimal objective  $\hat{\mu}^t = \mu (\hat{x}^t)$  if  $t\geq T_{\mathrm{Ave}}$  then Compute the CoRe threshold for the next iteration:  $\kappa^{t + 1} = \frac{\hat{\mu}^{t - T_{\mathrm{Ave}}} - \hat{\mu}^{t}}{T_{\mathrm{Ave}}};$  end   
end   
return  $D^{T_{\mathrm{MI}}},\hat{x}^{T_{\mathrm{MI}}},\mu (\hat{x}^{T_{\mathrm{MI}}})$
$\{X', y'\}$  (Step 11). The GP is updated with the updated training data  $\mathcal{D}^t = \{X^t, y^t\}$ , and its mean predictions at three points are fitted by a sinusoidal function (Step 14) for finding the new optimal point  $\hat{x}^t$  (Step 15). Before going to the next iteration, the CoRe threshold  $\kappa$  is updated according to the energy decrease in the last iterations (Step 19). In the early stage of the optimization, 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 steps.
F.3 EMICoRe Subroutine
The EMICoRe subroutine (Algorithm 3) receives the current optimal point  $\hat{\pmb{x}}$ , the current training data  $\{\pmb{X},\pmb{y}\}$ , the direction  $d$  to be explored, and the CoRe threshold  $\kappa$ , and returns the suggested observation points. Fixing the number of new observations per step to  $M = 2$ , we prepare pairs of candidate points (sampled on a grid) along the axis  $d$  as a candidate set  $\mathcal{C} = \{\check{\pmb{X}}^j\in \mathbb{R}^{D\times 2}\}_{j = 1}^{J_{\mathrm{SG}}(J_{\mathrm{SG}} - 1)}$  (Step 2).