SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Algorithm 3: EMICoRe subroutine
input :
$\hat{x}$  : current optimal point  $\mathcal{D} = \{\pmb {X},\pmb {y}\}$  : current training data  $d$  : direction to be explored  $\kappa$  : CoRe threshold   
params:  $M = 2:\#$  of suggested points  $J_{\mathrm{SG}}:\#$  of search grid points  $J_{\mathrm{OG}}:\#$  of evaluation grid points  $N_{\mathrm{MC}}:\#$  of Monte Carlo samples   
output :  $X^{\prime}$  : suggested observation points   
1 begin   
2 Prepare a candidate set  $\mathcal{C} = \{\check{X}^j\in \mathbb{R}^{D\times 2}\}_{j = 1}^{J_{\mathrm{SG}}(J_{\mathrm{SG}} - 1)};$    
3 /\* with  $J_{\mathrm{SG}}(J_{\mathrm{SG}} - 1)$  being the # of candidate pairs \*/   
4 for  $j = 1$  to  $J_{\mathrm{SG}}(J_{\mathrm{SG}} - 1)$ $/*\textit{j}$  denotes one pair of points \*/   
5 do   
6 Update GP adding the current candidate point  $\Rightarrow$ $\mathrm{GP}(\widetilde{\boldsymbol{X}})$  where  $\widetilde{\boldsymbol{X}} = (\boldsymbol {X},\widetilde{\boldsymbol{X}}^{j})$    
7 Compute the posterior variance  $s_{\widetilde{X}}(\pmb {x},\pmb {x})\forall \pmb{x}$  in the evaluation grid, along axis d;  $/*$  test GP uncertainty on evaluation points. \*/   
8 Find discrete approximation of the CoRe as  $\mathcal{Z}_{\widetilde{X}} = \{\pmb {x}\in \pmb{X}^{\mathrm{Grid}};s_{\widetilde{X}}(\pmb {x},\pmb {x})\leq \kappa^2\}$    
9 Use  $\hat{x}$  and  $X^{\mathrm{test}} = \mathcal{Z}_{\widetilde{X}}$  to compute mean and the covariance of GP posterior: p(f,ftest|X,y);   
10 Estimate acquisition function by quasi Monte Carlo sampling   
11  $a_{X}^{\mathrm{EMICoRe}}(j) = \frac{1}{M}\langle \max \{0,\hat{f} -\min (f^{\mathrm{test}})\} \rangle_{p(\hat{f},f^{\mathrm{test}}|X,y)}$    
12 end   
13 Find pair of observation points that maximizes EMICoRe   
14  $\hat{j} = \operatorname {argmax}_j a_X^{\mathrm{EMICoRe}}(j)$    
15 end   
16 return Suggested points to observe at next step  $X^{\prime} = \check{X}^{\hat{j}}$
For each candidate pair, the predictive variances of the updated GP are computed on  $\pmb{x} \in \pmb{X}^{\mathrm{test}}$ , points on a test grid, along the direction  $d$  (Step 6). This way, a discrete approximation of the CoRe is obtained by collecting the grid points where the variance is smaller than the threshold  $\kappa^2$  (Step 8). After computing the mean and the covariance of the current GP at the current optimum  $\hat{\pmb{x}}$  and on the (discrete) CoRe points (Step 9) — which is a  $\widetilde{D}$ -dimensional Gaussian for  $\widetilde{D} = |\hat{\pmb{x}} \cup \mathcal{Z}_{\widetilde{\pmb{X}}}| = 1 + |\mathcal{Z}_{\widetilde{\pmb{X}}}|$  — the EMICoRe acquisition function is estimated by quasi-Monte Carlo sampling (Step 11). After evaluating the acquisition functions for all candidate pairs of points, the best pair is returned as the suggested observation points (Step 16).
F.4 Parameter Setting
We automatically tune the sensitive parameters: the kernel smoothness parameter  $\gamma$  is optimized by maximizing the marginal likelihood in each iteration in the early phase of optimization, and at intervals in the later phase (see Appendix H for the concrete schedule in each experiment). The CoRe threshold  $\kappa$  is updated at every step and set to the average energy decrease of the last iterations as in Step 19 of Algorithm 2, which performs comparably to the best heuristic in our investigation below, see Table 1. The noise variance  $\sigma^2$  is set by observing  $f^{*}(\pmb{x})$  several times at several random
Table 1: Performance of EMICoRe depending on the choice of hyperparameters  $C_0$  and  $C_1$  for the CoRe threshold update rule (35). The best results are highlighted in bold, while  $\downarrow (\uparrow)$  indicates whether lower (higher) values are better.
Description	C0	C1	Energy ↓	Fidelity ↑
Default (Eq.(12))	0.0	1.0	-5.82 ± 0.14	0.85 ± 0.16
Extreme (small)	0.1	0.1	-5.82 ± 0.11	0.85 ± 0.16
High (large)	10.0	10.0	-5.72 ± 0.15	0.82 ± 0.16
Extreme (large)	10.0	100.0	-5.70 ± 0.16	0.80 ± 0.18
Best	0.1	10.0	-5.84 ± 0.09	0.87 ± 0.11
points and estimating  $\sigma^2 = \hat{\sigma}^{*2}(N_{\mathrm{shots}})$  before starting the optimization. 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 standard deviation  $\sigma_0$  is in the same order as the absolute value of the ground-state energy. The parameters  $T_{\mathrm{MI}},J_{\mathrm{SG}},J_{\mathrm{OG}}$  , and  $N_{\mathrm{MC}}$  should be set to sufficiently large values as long as the (classical) computation is tractable, and the performance is not sensitive to  $T_{\mathrm{NFT}}$  and  $T_{\mathrm{Ave}}$ . The parameter values used in our experiments are given in Appendix H.
Investigation of CoRe Threshold Update Rules: In our experiments in Section 4, the CoRe threshold is updated by Eq. (12). Here, we investigate whether a more fine-tuned update rule can improve the performance. Specifically, we test the following protocol:
$$
\kappa^ {t + 1} = \max  \left(C _ {0} \cdot \sigma , C _ {1} \cdot \frac {\hat {\mu} ^ {t - T _ {\mathrm {A v e}}} - \hat {\mu} ^ {t}}{T _ {\mathrm {A v e}}}\right), \tag {35}
$$
where  $C_0, C_1 \geq 0$  are the hyperparameters controlling the lower bound and the scaling of the average energy reduction, respectively. We note that  $\sigma$  is the standard deviation of the observation noise, and setting the hyperparameters to  $C_0 = 0$ ,  $C_1 = 1.0$  reduces to the default update rule (12). Table 1 shows the achieved energy and fidelity for different values of the hyperparameters  $C_0$  and  $C_1$ , after 600 observed points, in the setting of the Ising Hamiltonian at criticality and a  $(L = 3)$ -layered  $(Q = 5)$ -qubits quantum circuit with  $N_{\mathrm{shots}} = 1024$  readout shots. We observe that choosing  $C_0 = 0.1$  and  $C_1 = 10$  leads to the best performance; however, we also note that the setting used for the paper (12) achieves a similar performance.
Proof. We divide the proof into two steps.
G.1 Eq. (7)  $\Rightarrow$  Eq. (8)
The parameter shift rule (7) for  $a = 1 / 2$  gives
$$
2 \frac {\partial}{\partial x _ {d}} f ^ {*} (\boldsymbol {x}) = f ^ {*} \left(\boldsymbol {x} + \frac {\pi}{2} \boldsymbol {e} _ {d}\right) - f ^ {*} \left(\boldsymbol {x} - \frac {\pi}{2} \boldsymbol {e} _ {d}\right), \quad \forall \boldsymbol {x} \in [ 0, 2 \pi) ^ {D}, d = 1, \dots , D, \tag {36}
$$
which implies the differentiability of  $f^{*}(\pmb{x})$  in the whole domain  $[0, 2\pi)^{D}$ . For any  $d = 1, \dots, D$  and  $\hat{\pmb{x}} \in [0, 2\pi)^{D}$ , consider the one-dimensional subspace of the domain such that  $\mathcal{A}_{d,\hat{\pmb{x}}} = \{\hat{\pmb{x}} + \alpha \pmb{e}_d; \alpha \in [0, 2\pi)\}$ , and the following restriction of  $f^{*}(\cdot)$  on the subspace:
$$
\widetilde {f} _ {d, \hat {\boldsymbol {x}}} ^ {*} (\boldsymbol {x} _ {d}) \equiv f ^ {*} \big | _ {\mathcal {A} _ {d, \hat {\boldsymbol {x}}}} (\boldsymbol {x}) = f ^ {*} (\hat {\boldsymbol {x}} + (\boldsymbol {x} _ {d} - \hat {\boldsymbol {x}} _ {d}) \boldsymbol {e} _ {d}).
$$
For this restricted function, the parameter shift rule (36) applies as
$$
2 \frac {\partial}{\partial x _ {d}} \widetilde {f} _ {d, \hat {\mathbf {x}}} ^ {*} (x _ {d}) = \widetilde {f} _ {d, \hat {\mathbf {x}}} ^ {*} \left(x _ {d} + \frac {\pi}{2}\right) - \widetilde {f} _ {d, \hat {\mathbf {x}}} ^ {*} \left(x _ {d} - \frac {\pi}{2}\right), \quad \forall x _ {d} \in [ 0, 2 \pi), d = 1, \dots , D. \tag {37}
$$
The periodicity of  $f^{*}(\cdot)$  requires that  $\widehat{f}_{d,\widehat{\mathbf{x}}}^{*}(x_{d})$  can be written as a Fourier series,
$$
\widetilde {f} _ {d, \hat {\mathbf {x}}} \left(x _ {d}\right) = c _ {d, 0, 0} \left(\hat {\mathbf {x}} _ {\backslash d}\right) + \sum_ {\tau = 1} ^ {\infty} \left\{c _ {d, 1, \tau} \left(\hat {\mathbf {x}} _ {\backslash d}\right) \cos (\tau x _ {d}) + c _ {d, 2, \tau} \left(\hat {\mathbf {x}} _ {\backslash d}\right) \sin (\tau x _ {d}) \right\}, \tag {38}
$$
where  $\{c_{d,\cdot \cdot}(\hat{\pmb{x}}_{\backslash d})\}$  denote the Fourier coefficients, which depend on  $\hat{\pmb{x}}$  except for  $\hat{x}_d$ . Below, we omit the dependence of the Fourier coefficients on  $\hat{\pmb{x}}_{\backslash d}$  to avoid cluttering.
Substituting Eq. (38) into the left- and the right-hand sides of Eq. (37), respectively, gives
$$
2 \frac {\partial}{\partial x _ {d}} \widetilde {f} _ {d, \hat {\mathbf {x}}} ^ {*} (x _ {d}) = 2 \sum_ {\tau = 1} ^ {\infty} \tau \left(- c _ {d, 1, \tau} \sin (\tau x _ {d}) + c _ {d, 2, \tau} \cos (\tau x _ {d})\right), \tag {39}
$$
$$
\begin{array}{l} \widetilde {f} _ {d, \hat {\mathbf {x}}} ^ {*} \left(x _ {d} + \frac {\pi}{2}\right) - \widetilde {f} _ {d, \hat {\mathbf {x}}} ^ {*} \left(x _ {d} - \frac {\pi}{2}\right) = \sum_ {\tau = 1} ^ {\infty} \left(c _ {d, 1, \tau} \left\{\cos \left(\tau \left(x _ {d} + \frac {\pi}{2}\right)\right) - \cos \left(\tau \left(x _ {d} - \frac {\pi}{2}\right)\right) \right\} \right. \\ \left. + c _ {d, 2, \tau} \left\{\sin \left(\tau \left(x _ {d} + \frac {\pi}{2}\right)\right) - \sin \left(\tau \left(x _ {d} - \frac {\pi}{2}\right)\right) \right\}\right) \\ = 2 \sum_ {\tau = 1} ^ {\infty} \left(- c _ {d, 1, \tau} \sin (\tau x _ {d}) \sin \left(\frac {\tau \pi}{2}\right) + c _ {d, 2, \tau} \cos (\tau x _ {d}) \sin \left(\frac {\tau \pi}{2}\right)\right) \\ = 2 \sum_ {\tau = 1} ^ {\infty} \sin \left(\frac {\tau \pi}{2}\right) \left(- c _ {d, 1, \tau} \sin (\tau x _ {d}) + c _ {d, 2, \tau} \cos (\tau x _ {d})\right). \tag {40} \\ \end{array}
$$
Since Eq. (37) requires that Eqs. (39) and (40) are equal to each other for any  $x_{d} \in [0,2\pi)$  and  $d = 1,\dots ,D$ , it must hold that
$$
\tau = \sin \left(\frac {\tau \pi}{2}\right) \quad \forall \tau \quad \text {s u c h} c _ {d, 1, \tau} \neq 0 \text {o r} c _ {d, 2, \tau} \neq 0. \tag {41}
$$
Since Eq. (41) can hold only for  $\tau = 1$ , we deduce that
$$
c _ {d, 1, \tau} = c _ {d, 2, \tau} = 0, \quad \forall \tau \neq 1, d = 1, \dots , D.
$$
Therefore, the restricted function must be the first-order sinusoidal function:
$$
\widetilde {f} _ {d, \hat {\boldsymbol {x}}} (x _ {d}) = c _ {d, 0, 0} (\hat {\boldsymbol {x}} _ {\backslash d}) + c _ {d, 1, 1} (\hat {\boldsymbol {x}} _ {\backslash d}) \cos (x _ {d}) + c _ {d, 2, 1} (\hat {\boldsymbol {x}} _ {\backslash d}) \sin (x _ {d}). \tag {42}
$$
As the most general function form that satisfies Eq. (42) for all  $d = 1, \ldots, D$ , we have
$$
\begin{array}{l} f ^ {*} (\boldsymbol {x}) = \sum_ {\boldsymbol {\xi} \in \{0, 1, 2 \} ^ {D}} \widetilde {b} _ {\boldsymbol {\xi}} \prod_ {d = 1} ^ {D} 1 ^ {\mathbb {1} (\xi_ {d} = 0)} \cdot (\cos x _ {d}) ^ {\mathbb {1} (\xi_ {d} = 1)} \cdot (\sin x _ {d}) ^ {\mathbb {1} (\xi_ {d} = 2)} \\ = \boldsymbol {b} ^ {\top} \cdot \mathbf {v e c} \left(\otimes_ {d = 1} ^ {D} (1, \cos x _ {d}, \sin x _ {d}) ^ {\top}\right). \\ \end{array}
$$
Here,  $\pmb{\xi} \in \{0,1,2\}^D$  takes the value of either 0, 1, or 2, specifying the dependence on  $x_d$  —constant, cosine, or sine—for each entry, and  $\widetilde{\pmb{b}} = (\widetilde{b}_{\pmb{\xi}})_{\pmb{\xi} \in \{0,1,2\}^D}$  is the  $3^D$ -dimensional coefficient vector indexed by  $\pmb{\xi}$ . With the appropriate bijective mapping  $\iota : \{0,1,2\}^D \mapsto 1, \dots, 3^D$  consistent with the definition of the vectorization operator  $\mathbf{vec}(\cdot)$ , we defined  $\pmb{b} \in \mathbb{R}^{3^D}$  such that  $b_{\iota(\pmb{\xi})} = \widetilde{b}_{\pmb{\xi}}$ .  $\mathbb{1}(\cdot)$  is the indicator function, which is equal to one if the event is true and zero otherwise.
G.2 Eq. (8)  $\Rightarrow$  Eq. (7)
For any  $f^{*}(\cdot)$  in the form of Eq. (8), the left- and right-hand sides of the parameter shift rule (36) can be, respectively, written as
$$
2 \frac {\partial}{\partial x _ {d}} f ^ {*} (\boldsymbol {x}) = 2 \left(- c _ {d, 1, 1} \left(\boldsymbol {x} _ {\backslash d}\right) \sin \left(x _ {d}\right) + c _ {d, 2, 1} \left(\boldsymbol {x} _ {\backslash d}\right) \cos \left(x _ {d}\right)\right), \tag {43}
$$
$$
\begin{array}{l} f ^ {*} \left(\boldsymbol {x} + \frac {\pi}{2} \boldsymbol {e} _ {d}\right) - f ^ {*} \left(\boldsymbol {x} - \frac {\pi}{2} \boldsymbol {e} _ {d}\right) = c _ {d, 1, 1} (\boldsymbol {x} _ {\backslash d}) \left\{\cos \left(x _ {d} + \frac {\pi}{2}\right) - \cos \left(x _ {d} - \frac {\pi}{2}\right) \right\} \\ + c _ {d, 2, 1} \left(\boldsymbol {x} _ {\backslash d}\right) \left\{\sin \left(x _ {d} + \frac {\pi}{2}\right) - \sin \left(x _ {d} - \frac {\pi}{2}\right) \right\} \\ = 2 \left(- c _ {d, 1, 1} (\boldsymbol {x} _ {\backslash d}) \sin (x _ {d}) \sin \left(\frac {\pi}{2}\right) + c _ {d, 2, 1} (\boldsymbol {x} _ {\backslash d}) \cos (x _ {d}) \sin \left(\frac {\pi}{2}\right)\right) \\ = 2 \left(- c _ {d, 1, 1} \left(\boldsymbol {x} _ {\backslash d}\right) \sin \left(x _ {d}\right) + c _ {d, 2, 1} \left(\boldsymbol {x} _ {\backslash d}\right) \cos \left(x _ {d}\right)\right), \tag {44} \\ \end{array}
$$
which coincide with each other. This completes the proof.
Table 2: Choice of coupling parameters for the Ising and Heisenberg Hamiltonians for reproducing the experiments in Section 4 and Appendix I.
Ising	Heisenberg
--j-couplings (JX, JY, JZ)	(-1.0, 0.0, 0.0)	(1.0, 1.0, 1.0)
--h-couplings (hX, hY, hZ)	(0.0, 0.0, -1.0)	(1.0, 1.0, 1.0)
Table 3: Additional non-default parameters for reproducing the experiments in Section 4.1.
Command	Values
--hyperopt	optim=grid,steps=80,interval=75*1+100*25,loss=m11
--kernel-params	sigma_0=1.0,gamma=2.0
Every numerical experiment, unless stated otherwise, consists of 50 independent seeded trials. Every seed (trial) starts with one datapoint,  $\mathcal{D}^0 = (\hat{\pmb{x}}^0,\hat{y}^0)$ , with  $\pmb{x}^0$  being an initial point uniformly drawn from  $[0,2\pi)^D$ , and  $y^0$  being the associated energy at  $\pmb{x}^0$  evaluated on the quantum computer. Those 50 initial pairs are cached and, when an optimization trial starts with a given seed, the corresponding cached initial pair is loaded. This allows a fair comparison of different optimization methods: all methods start from the same set of initialization points.
The Qiskit [43] open-source library is used to classically simulate the quantum computer, whereas the rest of the implementation uses pure Python. All numerical experiments have been performed on Intel Xeon Silver 4316 @ 2.30GHz CPUs, and the code with instructions on how to run and reproduce the results is publicly available on GitHub [44].
VQE kernel analysis (Section 4.1): We compare the performance of our proposed VQE-kernel for VQE with the Ising Hamiltonian, i.e., the Heisenberg Hamiltonian (13) with the coupling parameters set to
$$
J _ {X} = - 1, J _ {Y} = 0, J _ {Z} = 0, \quad h _ {X} = 0, h _ {Y} = 0, h _ {Z} = - 1 \tag {45}
$$
and open boundary conditions (see Table 2). For the variational quantum circuit  $G(\pmb{x})$ , we use a  $(Q = 3)$ -qubit,  $(L = 3)$ -layered Efficient SU(2) circuit. In this case, the search domain is  $\pmb{x} \in [0,2\pi)^{24}$ , according to Eq. (31). The number of readout shots is set to  $N_{\mathrm{shots}} = 1024$ . The baseline kernels are the Gaussian-RBF kernel,
$$
k ^ {\mathrm {R B F}} \left(\boldsymbol {x}, \boldsymbol {x} ^ {\prime}\right) = \sigma_ {0} ^ {2} \exp \left(- \frac {\| \boldsymbol {x} - \boldsymbol {x} ^ {\prime} \| ^ {2}}{2 \gamma^ {2}}\right), \tag {46}
$$
and the Periodic kernel [67],
$$
k ^ {\text {p e r i o d}} \left(\boldsymbol {x}, \boldsymbol {x} ^ {\prime}\right) = \sigma_ {0} ^ {2} \exp \left(- \sum_ {d = 1} ^ {D} \frac {1}{2 \gamma^ {2}} \sin^ {2} \left(\frac {x _ {d} - x _ {d} ^ {\prime}}{2}\right)\right), \tag {47}
$$
which are compared with our VQE kernel (9) in terms of the standard BO performance in Figure 2. Each kernel has two hyperparameters, the prior variance  $\sigma_0^2$  and the smoothness parameter  $\gamma$ . For all three kernels, the prior variance is fixed to  $\sigma_0^2 = 1$ , and the smoothness parameter  $\gamma$  is automatically tuned by marginal likelihood maximization (grid search) in each iteration in the early stage ( $t = 0, \dots, 75$ ), and after every 100 iterations in the later stage ( $t = 76, \dots$ ).
For the standard BO, we used the EI acquisition function, which is maximized by L-BFGS [45]. In the code, the SciPy [68] implementation of L-BFGS was used and all experiments were run using the same default parameter set in the code. Detailed commands for reproducing the results can be found in Table 3.
Table 4: Standard choice of EMICoRe hyperparameters for experiments in Section 4.2 and Appendix I (unless specified otherwise).
General params
--n-qbits	{3,5,7}	# of qubits
--n-layers	{3,3,5}	# of circuit layers
--circuit	esu2	Circuit name
--pbc	False	Open Boundary Conditions
--n-readout (TN)	{100,300,500}	# iterations for BO
--n-iter (TMI)	{100,300,500}	# iterations for BO
--kernel	vqe	Name of the kernel
--hyperopt	Hyperparams optimization
optim	grid	Grid-search optimization of γ
max_gamma	20	Max value for γ
interval	100*1+20*9+10*100	Scheduling for grid-search
steps	120	# steps in grid
loss	mll	Loss type
--acq-params	EMICoRe params
func	func=ei	Base acq. func. type
optim	optim=emicore	Optimizer type
pairsize (JSG)	20	# of candidate points
gridsize (JOG)	100	# of evaluation points
corethresh (κ)	1.0	CoRe threshold κ
corethresh_width (T_Ave)	10	# averaging steps to update κ
coremin_scale (C0)	0.0	coefficient C0 in eq. (35)
corethresh_scale (C1)	1.0	coefficient C1 in eq. (35)
samplesize (NMC)	100	# of MC samples
smo-steps (TNFT)	0	# of initial NFT steps
smo-axis	True	Sequential direction choice
EMICoRe analysis (Section 4.2): In this experiment, we compare the optimization performance of our NFT-with-EMICoRe with the NFT baselines (sequential and random) on VQE with both the Ising Hamiltonian, for which the coupling parameters are given in Eq. (45), and the Heisenberg Hamiltonian, for which the coupling parameters are set to
$$
J _ {X} = 1, J _ {Y} = 1, J _ {Z} = 1 \quad h _ {X} = 1, h _ {Y} = 1, h _ {Z} = 1. \tag {48}
$$
For the variational quantum circuit  $G(\pmb{x})$ , we use a  $(Q = 5)$ -qubit,  $(L = 4)$ -layered Efficient SU(2) circuit with open boundary conditions, giving  $(D = 40)$ -dimensional search domain. The number of readout shots is set to  $N_{\mathrm{shots}} = 1024$  in the experiment shown in Figure 3. In the same format as Figure 3, Figures 9-17 in Appendix I.3 compare EMICoRe with the baselines for different setups of  $Q$ ,  $L$ , and  $N_{\mathrm{shots}}$ .
For NFT-with-EMICoRe (Algorithm 2 and Algorithm 3), where the VQE kernel is used, the prior standard deviation is set to a value roughly proportional to the number of qubits  $Q$ ; specifically for  $Q = 5$  we set  $\sigma_0 = 6$ . The smoothness parameter  $\gamma$  is automatically tuned by marginal likelihood maximization in each iteration in the early stage  $(t = 0, \dots, 100)$ , after every 9 iterations in the middle phase  $(t = 101, \dots, 280)$ , and after every 100 iterations in the last phase  $(t = 281, \dots)$ . The other parameters are set to  $T_{\mathrm{MI}} = 300$ ,  $J_{\mathrm{SG}} = 20$ ,  $J_{\mathrm{OG}} = 100$ ,  $N_{\mathrm{MC}} = 100$ ,  $T_{\mathrm{NFT}} = 0$  and  $T_{\mathrm{Ave}} = 10$ . All relevant hyperparameters are collected in Table 4 along with the corresponding flags in our code.
The command options that specify the VQE setting, the kernel optimization schedule, and the other parameter settings for EMICoRe, are summarized in Table 2 and Table 4.
EMICoRe (ours) NFT-Sequential NFT-Random
Figure 7: Energy (left) and fidelity (right) for EMICoRe (ours) and the baselines, NFT-sequential and NFT-random, up to 6000 observations. Results are for the Ising Hamiltonian with a  $(L = 3)$ -layered  $(Q = 5)$ -qubits quantum circuit and  $N_{\mathrm{shots}} = 1024$ .
Table 5: Energy and fidelity achieved after 6000 observations in the experiment in Figure 7. The best results are highlighted in bold, while  $\downarrow (\uparrow)$  indicates lower (higher) values are better.
Algorithm	Energy ↓	Fidelity ↑
EMICoRe (ours)	-5.97 ± 0.05	0.98 ± 0.04
NFT-random	-5.92 ± 0.08	0.92 ± 0.09
NFT-sequential	-5.93 ± 0.09	0.92 ± 0.16
Here, we report on additional experimental results.
I.1 Convergence to Ground State
The experiments from Section 4.2, compared the performance between our EMICoRe and the NFT baselines up to 600 observed points, where the optimization has not yet converged. Here, we perform a longer optimization with up to 6000 observations to confirm the ability of EMICoRe to converge to the ground state. In Figure 7 the energy and fidelity plots show the optimization progress for the Ising model with a  $(L = 3)$ -layered  $(Q = 5)$ -qubits quantum circuit and  $N_{\mathrm{shots}} = 1024$  readout shots. The portrait plot on the right shows the distribution over 50 independent trials of the final solutions after 6000 observations. The mean and the standard deviation of the achieved energy and fidelity are summarized in Table 5. We observe that EMICoRe achieves an average fidelity above  $95\%$  after 1000 observations, and reaches  $98\%$  fidelity at 4000 observations. In contrast, the NFT baselines require all 6000 observations in order to achieve a fidelity of  $92\%$ , exhibiting a much slower convergence. This result confirms that EMICoRe robustly converges to the ground state, independent of the individual trial's initialization.
Note that the GP regression exhibits cubic computational complexity with respect to the number of samples, thus significantly slowing down the optimization process with thousands of observed points. As a remedy, we limit the number of utilized samples by discarding old observations, i.e., we choose inducing points for the GP based on the chronological order of observations. We found in preliminary experiments that choosing the last 100 observations as inducing points is sufficient to achieve good results. In the experiments above (see Figure 7), we keep the number of inducing points above 100 and below 120, where we discard the 20 oldest points when exceeding a number of 120 observations. This strategy is implemented in our public code [44] through the option --inducer last_slack:retain=100:slack=20, where last_slack indicates the criterion to choose the inducing points, and where retain and slack can be used to specify the minimum number of points retained and the number of samples that can be observed beyond the minimum, before discarding. Hence, the model discards slack=20 observations when the number of observed points equals to the sum of the two, i.e., slack + retained.
(a) Ising off-criticality
(b) Ising at criticality
Figure 8: Energy (left) and fidelity (right) for EMICoRe (ours) compared to three different baselines: NFT-sequential, NFT-random, and EI with VQE kernel. Results for the Ising Hamiltonian off-criticality and at criticality are shown in the top and bottom rows respectively.
I.2 Ising Hamiltonian Off-Criticality
In Section 4, we focused on the Ising and Heisenberg Hamiltonians with the parameters,  $J = (J_{X}, J_{Y}, J_{Z})$  and  $h = (h_{X}, h_{Y}, h_{Z})$  in Equation (13), set to criticality in the thermodynamic limit. Such choices are expected to be most challenging for the VQE optimization because the corresponding ground states tend to be highly entangled due to the quantum phase transition. As an ablation study, we here conduct experiments for an off-critical setting. Specifically, we evaluate the optimization performance for the Ising Hamiltonian off-criticality  $\{J = (0, 0, -1); h = (1.5, 0, 0)\}$ . Figure 8 (top) shows the energy (left) and the fidelity (right) achieved by EMICoRe (ours), NFT-sequential, NFT-random, and EI with VQE kernel after 600 iterations for a  $(L = 3)$ -layered  $(Q = 5)$ -qubits quantum circuit with  $N_{\mathrm{shots}} = 1024$  readout shots. For comparison, we also show in Figure 8 (bottom) the performance for the Ising Hamiltonian at criticality  $\{J = (-1, 0, 0); h = (0, 0, -1)\}$ . We observe that the off-criticality setting is significantly easier, while the plain BO with EI (without EMICoRe) falls somewhat short, although closely behind NFT-random at 600 observed points.
I.3 Different Setups for Qubits and Layers