SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Kim A. Nicoli $^{1,2,3*}$  Christopher J. Anders $^{3,4*}$  Lena Funcke $^{1,2}$  Tobias Hartung $^{5,6}$  Karl Jansen $^{7}$  Stefan Kuhn $^{7}$  Klaus-Robert Muller $^{3,4,8,9}$  Paolo Stornati $^{10}$  Pan Kessel $^{11}$  Shinichi Nakajima $^{3,4,12}$
$^{1}$ Transdisciplinary Research Area (TRA) Matter, University of Bonn, Germany  $^{2}$ Helmholtz Institute for Radiation and Nuclear Physics (HISKP)
$^{3}$ Berlin Institute for the Foundations of Learning and Data (BIFOLD)
$^{4}$  Technische Universität Berlin, Germany,  $^{5}$  Northeastern University London, UK  
 $^{6}$  Khoury College of Computer Sciences, Northeastern University, USA
$^{7}$  CQTA, Deutsches Elektronen-Synchrotron (DESY), Zeuthen, Germany
$^{8}$  Department of Artificial Intelligence, Korea University, Korea
9Max Planck Institut für Informatik, Saarbrücken, Germany
$^{10}$ Institute of Photonic Sciences, The Barcelona Institute of Science and Technology (ICFO)
11Prescient Design, gRED, Roche, Switzerland, 12RIKEN Center for AIP, Japan
In this paper, we propose a novel and powerful method to harness Bayesian optimization for Variational Quantum Eigensolvers (VQEs)—a hybrid quantum-classical protocol used to approximate the ground state of a quantum Hamiltonian. Specifically, we derive a VQE-kernel which incorporates important prior information about quantum circuits: the kernel feature map of the VQE-kernel exactly matches the known functional form of the VQE's objective function and thereby significantly reduces the posterior uncertainty. Moreover, we propose a novel acquisition function for Bayesian optimization called Expected Maximum Improvement over Confident Regions (EMICoRe) which can actively exploit the inductive bias of the VQE-kernel by treating regions with low predictive uncertainty as indirectly "observed". As a result, observations at as few as three points in the search domain are sufficient to determine the complete objective function along an entire one-dimensional subspace of the optimization landscape. Our numerical experiments demonstrate that our approach improves over state-of-the-art baselines.
Quantum computing raises the exciting future prospect to efficiently tackle currently intractable problems in quantum chemistry [1], many-body physics [2], combinatorial optimization [3], machine learning [4], and beyond. Rapid progress over the last years has led to the development of the first noisy intermediate-scale quantum (NISQ) devices [5]; quantum hardware with several hundreds of qubits that can be harnessed to outperform classical computers on specific tasks (see, e.g., Ref. [6]). However, these tasks have been specifically designed to be hard on classical computers and easy on quantum computers, while being of no practical use. As we have entered the NISQ era, one of the grand challenges of quantum computing lies in the development of algorithms for NISQ devices that may exhibit a quantum advantage on a task of practical relevance.
*These authors contributed equally to the work. Correspondence to knicoli@uni-bonn.de and {anders,nakajima}@tu-berlin.de
One promising approach toward using NISQ devices is hybrid quantum-classical algorithms, such as VQEs [7, 8], which can be used to compute ground states of quantum Hamiltonians. VQEs can be seen as the quantum counterpart of neural networks: while classical neural networks model functions by parametric layers, the parametric quantum circuits used in VQEs represent variational wave functions by parametric quantum gates acting on qubits. During training, we aim to find a suitable choice for the variational parameters of the wave function such that the quantum mechanical expectation value of the Hamiltonian is minimized. From the optimization perspective, the quantum mechanical nature of the energy measurement is of little relevance. The training of VQEs can thus be regarded as a specific, albeit particularly challenging, noisy black-box optimization problem. Namely, we solve
$$
\min  _ {\boldsymbol {x} \in [ 0, 2 \pi) ^ {D}} f ^ {*} (\boldsymbol {x}), \tag {1}
$$
where  $f^{*}(\pmb{x})$  is estimated from costly noisy observation,  $y = f^{*}(\pmb{x}) + \varepsilon$ , on the quantum device.
Efficiently exploring the energy landscape of the VQE is crucial for successful optimization. This requires leveraging strong prior knowledge about the VQE objective function  $f^{*}(\cdot)$ . For instance, gradient-based optimization methods commonly utilize the parameter shift rule, which takes advantage of the specific structure of variational quantum circuits to compute gradients efficiently [9, 10]. Another approach is the Nakanishi-Fuji-Todo (NFT) method [11], a version of sequential minimal optimization (SMO) [12], which solves sub-problems sequentially. Nakanishi et al. [11] derived the explicit functional form of the VQE objective, enabling coordinate-wise global optimization with only two observations per iteration.
Given the highly non-trivial nature of classical optimization in VQE, machine learning represents an appealing tool to leverage the informative underlying physics toward a more effective search for a global optimum. Bayesian Optimization (BO) [13, 14] has been recently applied to VQE [15]. These methods have the distinct advantage that they can take the inherently noisy nature of the NISQ circuits into account. Unfortunately, these methods are yet to be competitive, especially in the high dimensional regime, because of their poor scalability and overemphasized-exploration behavior [16].
To overcome this limitation, this paper introduces a novel and powerful method for BO of VQEs, capitalizing on VQE-specific properties as physics-informed prior knowledge. To this end, we propose a VQE-kernel, designed with feature vectors that precisely align with the basis functions of the VQE objective. This incorporation of a strong inductive bias maximizes the statistical efficiency of Gaussian process (GP) regression and guarantees that GP posterior samples reside within the VQE function space. To further harness this powerful inductive bias, we present a novel acquisition function named Expected Maximum Improvement over Confident Regions (EMICoRe). EMICoRe operates by initially predicting the posterior variance and treating the points with low posterior variances as "observed" points. Subsequently, these indirectly observed points, alongside the directly observed ones, form the Confident Regions (CoRe) on which safe optimization of the GP mean is conducted to determine the current optimum. EMICoRe evaluates the expected maximum improvement of the best points in CoRe before and after the candidate points are observed. By utilizing EMICoRe, our approach combines the strengths of NFT [11] and BO, complementing each other in a synergistic manner. BO enhances the efficiency of NFT by replacing sub-optimal deterministic choices of next observation points, while the NFT procedure significantly constrains the exploration space of BO, leading to remarkable improvements in scalability.
Our numerical experiments demonstrate the performance gains of using the VQE kernel, and significant improvement of our NFT-with-EMICoRe approach in VQE problems with different Hamiltonians, different numbers of qubits, etc. As an additional contribution, we prove that two known important properties of VQE, the parameter shift rule [9] and the sinusoidal function-form [11], are equivalent, implying that they are not two different properties but two expressions of a single property.
Related Work Numerous optimization methods for the VQE protocol have been proposed: gradient-based methods often rely on the parameter shift rule [9, 10], while NFT harnesses the specific functional form of the VQE objective [11] to establish SMO. BO has also been applied for VQE minimization [15]. Therein, it was shown that combining periodic kernel [17] and noisy expected improvement acquisition function [18] improves the performance of BO over the standard RBF kernel, and can make BO comparable to the state-of-the-art methods in the regime of small qubits and high observation noise.
Figure 1: Illustration of EMICoRe (ours) and NFT [11] (baseline) procedures. In each step of NFT (bottom row), a) given the current best point  $\hat{\pmb{x}}^t$  and the direction  $d^{t}$  to be explored, b) next observation points are chosen in a deterministic fashion. Then, c) the minimum along the line is found by the sinusoidal function fitting to the three points, and d) the next optimization step for the new direction  $d^{t + 1}$  starts from the found minimum  $\hat{\pmb{x}}^{t + 1} = \hat{\pmb{x}}_{\mathrm{min}}^t$ . The EMICoRe procedure (top row) uses GP regression and BO in the steps highlighted by the light blue box: b) the next observation points are chosen by BO with the EMICoRe acquisition function based on the GP trained on the previous observations, and c) minimizing the predictive mean function of the updated GP with the new observations gives the best point  $\hat{\pmb{x}}_{\mathrm{min}}^t$ .
BO is a versatile tool for black-box optimization in many applications [19, 20, 21, 22] including hyperparameter tuning of deep neural networks [23]. Most work on BO uses GP regression, and computes an acquisition function whose maximizer is suggested as the next observation point. Many acquisition functions, including lower confidence bound [24, 25], probability of improvement [26], Expected Improvement (EI) [27], entropy search [28, 29], and knowledge gradient [30] have been proposed. The most common acquisition function is EI, of which many generalizations exist: noisy EI (NEI) [18, 31] for dealing with observation noise, parallel EI [31] for batch sample suggestions, and EI per cost [23] for cost sensitivity. Our EMICoRe acquisition function is a generalization of noisy EI and parallel EI with the key novelty of introducing CoRe, which defines the indirectly observed points. Note the difference between the trust region [16] and CoRe: based on the predictive uncertainty, the former restricts the region to be explored, while the latter expands the observed points.
2.1 Bayesian Optimization
Let  $f^{*}(\cdot):\mathcal{X}\mapsto \mathbb{R}$  be an unknown (black-box) objective function to be minimized. BO [13, 14] approximates the objective with a surrogate function  $f(\pmb {x})\approx f^{*}(\pmb {x})$ , and suggests promising points to be observed in each step, such that the objective is likely to be improved considerably. A commonly used surrogate is the Gaussian process (GP) regression model [32] with one-dimensional Gaussian likelihood and GP prior,
$$
p (y | \boldsymbol {x}, f (\cdot)) = \mathcal {N} _ {1} (y; f (\boldsymbol {x}), \sigma^ {2}), \quad p (f (\cdot)) = \mathrm {G P} (f (\cdot); \nu (\cdot), k (\cdot , \cdot)), \tag {2}
$$
which is trained on the (possibly) noisy observations  $y = f^{*}(\pmb{x}) + \varepsilon$  made in the previous iterations. Here,  $\sigma^2$  is the variance of observation noise  $\varepsilon$ , and  $\nu(\cdot)$  and  $k(\cdot, \cdot)$  are the prior mean and the kernel (covariance) functions, respectively. Throughout the paper, we set the prior mean function to be zero, i.e.,  $\nu(\pmb{x}) = 0, \forall \pmb{x} \in \mathcal{X}$ . Let  $\{\pmb{X}, \pmb{y}\}$  be  $N$  training samples, where  $\pmb{X} = (\pmb{x}_1, \dots, \pmb{x}_N) \in \mathcal{X}^N$  and  $\pmb{y} = (y_1, \dots, y_N)^\top \in \mathbb{R}^N$ . Since the GP prior is conjugate for the Gaussian likelihood<sup>II</sup>, the posterior of the GP regression model (2) is also a GP, i.e.,  $p(f(\cdot)|\pmb{X}, \pmb{y}) = \mathrm{GP}(f(\cdot); \mu_{\pmb{X}}(\cdot), s_{\pmb{X}}(\cdot, \cdot))$ , and thus, for arbitrary  $M$  test inputs  $\pmb{X}' = (\pmb{x}'_1, \dots, \pmb{x}'_M) \in \mathcal{X}^M$ , the posterior of the function values  $f' = (f(\pmb{x}_1'), \dots, f(\pmb{x}_M'))^\top \in \mathbb{R}^M$  is the  $M$ -dimensional Gaussian with its mean and covariance
See Chap. 2 of Ref. [33] for the conjugacy in Bayesian inference.
analytically given as
$$
p \left(\boldsymbol {f} ^ {\prime} \mid \boldsymbol {X}, \boldsymbol {y}\right) = \mathcal {N} _ {M} \left(\boldsymbol {f} ^ {\prime}; \boldsymbol {\mu} _ {\boldsymbol {X}} ^ {\prime}, \boldsymbol {S} _ {\boldsymbol {X}} ^ {\prime}\right), \quad \text {w h e r e} \tag {3}
$$
$$
\boldsymbol {\mu} _ {X} ^ {\prime} = \boldsymbol {K} ^ {\prime \top} \left(\boldsymbol {K} + \sigma^ {2} \boldsymbol {I} _ {N}\right) ^ {- 1} \boldsymbol {y}, \quad \boldsymbol {S} _ {X} ^ {\prime} = \boldsymbol {K} ^ {\prime \prime} - \boldsymbol {K} ^ {\prime \top} \left(\boldsymbol {K} + \sigma^ {2} \boldsymbol {I} _ {N}\right) ^ {- 1} \boldsymbol {K} ^ {\prime}. \tag {4}
$$
Here,  $\pmb{K} = k(\pmb{X},\pmb{X})\in \mathbb{R}^{N\times N}$ ,  $\pmb{K}^{\prime} = k(\pmb{X},\pmb{X}^{\prime})\in \mathbb{R}^{N\times M}$ , and  $\pmb{K}^{\prime \prime} = k(\pmb{X}^{\prime},\pmb{X}^{\prime})\in \mathbb{R}^{M\times M}$  are the train, train-test, and test kernel matrices, respectively, where  $k(\pmb {X},\pmb{X}^{\prime})$  denotes the kernel matrix evaluated at each column of  $\pmb{X}$  and  $\pmb{X}^{\prime}$  such that  $(k(\pmb {X},\pmb{X}^{\prime}))_{n,m} = k(\pmb{x}_n,\pmb{x}_m)$ . Moreover,  $\pmb{I}_N\in \mathbb{R}^{N\times N}$  denotes the identity matrix, and the subscript  $\pmb{X}$  of posterior means and covariances specifies the input points on which the GP was trained (see Appendix B for details of GP regression).<sup>III</sup>
In the general BO framework [14],  $M \geq 1$  input points to be observed next are chosen by (approximately) solving the following maximization problem in each iteration:  $\max_{\boldsymbol{X}'} a_{\boldsymbol{X}^{t-1}}(\boldsymbol{X}')$ , where  $\boldsymbol{X}^t$  denotes the training input points observed until the  $t$ -th iteration, and  $a_{\boldsymbol{X}}(\cdot)$  is an acquisition function computed based on the GP trained on the observations  $\boldsymbol{y}$  at  $\boldsymbol{X}$ . The acquisition function evaluates the promising-ness of the new input points  $\boldsymbol{X}'$ , and should therefore give a high value if observing  $\boldsymbol{X}'$  likely improves the current best score considerably. Common choices for the acquisition function are EI [27, 34],  $a_{\boldsymbol{X}}^{\mathrm{EI}}(\boldsymbol{x}') = \left\langle \max(0, \underline{f} - f') \right\rangle_{p(f'|_{\boldsymbol{X}}, \boldsymbol{y})}$ , and its variants. Here,  $\underline{f}$  denotes the current best observation, i.e.,  $\underline{f} = \min_{n \in \{1, \dots, N\}} f(\boldsymbol{x}_n)$ , and  $\langle \cdot \rangle_p$  denotes the expectation value with respect to the distribution  $p$ . EI covers the case where the observation noise is negligible and only one sample is chosen in each iteration, i.e.,  $\sigma^2 \ll 1$ ,  $M = 1$ , and its analytic solution makes EI handy (see Appendix B). In the general case where  $\sigma^2 > 0$ ,  $M \geq 1$ , generalizations of EI should be considered, such as NEI [18, 31],
$$
a _ {\boldsymbol {X}} ^ {\mathrm {N E I}} \left(\boldsymbol {X} ^ {\prime}\right) = \left\langle \max  (0, \min  (\boldsymbol {f}) - \min  \left(\boldsymbol {f} ^ {\prime}\right)) \right\rangle_ {p \left(\boldsymbol {f}, \boldsymbol {f} ^ {\prime} \mid \boldsymbol {X}, \boldsymbol {y}\right)}, \tag {5}
$$
which appropriately takes into account the correlation between observations. NEI is estimated by quasi Monte Carlo sampling, and its maximization is approximately performed by sequentially adding a point to  $\pmb{X}^{\prime}$  until  $M$  points are collected.
2.2 Variational Quantum Eigensolver (VQE)
The VQE [7, 8] is a hybrid quantum-classical computing protocol for estimating the ground-state energy of a given quantum Hamiltonian for a  $Q$ -qubit system. The quantum computer is used to prepare a parametric quantum state  $|\psi_{\mathbf{x}}\rangle$ , which depends on  $D$  angular parameters  $\mathbf{x} \in \mathcal{X} = [0,2\pi)^{D}$ . This trial state  $|\psi_{\mathbf{x}}\rangle$  is generated by applying  $D'(\geq D)$  quantum gate operations,  $G(\mathbf{x}) = G_{D'} \circ \dots \circ G_1$ , to an initial quantum state  $|\psi_0\rangle$ , i.e.,  $|\psi_{\mathbf{x}}\rangle = G(\mathbf{x})|\psi_0\rangle$ . All gates  $\{G_{d'}\}_{d'=1}^{D'}$  are unitary, and we assume in this paper that  $x_d$  parametrizes only a single gate  $G_{d'(d)}(x_d)$ , where  $d'(d)$  specifies the gate parametrized by  $x_d$ . Thus,  $D$  of the  $D'$  gates are parametrized by each entry of  $\mathbf{x}$  exclusively. We consider parametric gates of the form  $G_{d'}(x) = U_{d'}(x) = \exp(-ixP_{d'} / 2)$ , 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 the single-qubit gates such as  $R_X(x) = \exp(-i\theta \sigma_q^X)$ , but also the 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  $q_1 \neq q_2$  which are commonly realized in trapped-ion quantum hardwares [35, 36].
The quantum computer evaluates the energy of the resulting quantum state  $|\psi_{\pmb{x}}\rangle$  by observing
$$
y = f ^ {*} (\boldsymbol {x}) + \varepsilon , \quad \text {w h e r e} \quad f ^ {*} (\boldsymbol {x}) = \left\langle \psi_ {\boldsymbol {x}} \right| H \left| \psi_ {\boldsymbol {x}} \right\rangle = \left\langle \psi_ {0} \right| G (\boldsymbol {x}) ^ {\dagger} H G (\boldsymbol {x}) \left| \psi_ {0} \right\rangle \tag {6}
$$
and  $\dagger$  denotes the Hermitian conjugate. The observation noise  $\varepsilon$  in our numerical experiments will only incorporate shot noise, and we will not consider the hardware-dependent errors induced by imperfect qubits, gates, and measurements. For each observation, multiple readout shots  $N_{\mathrm{shots}}$  are acquired to suppress the variance  $\sigma^{*2}(N_{\mathrm{shots}})$  of the noise  $\varepsilon$ . Since the observation  $y$  is the sum of many random variables, it approximately follows a Gaussian distribution, according to the central limit theorem. The Gaussian likelihood in the GP regression model (2) therefore approximates the observation  $y$  well if  $f(\pmb{x}) \approx f^{*}(\pmb{x})$  and  $\sigma^2 \approx \sigma^{*2}(N_{\mathrm{shots}})$ . With the quantum computer that
Note that the subscript does not necessarily specify all dependencies. For example,  $\pmb{\mu}_{\pmb{X}}^{\prime}$  also depends on  $\pmb{y}$ . In Appendix E, we discuss how the theory and our method can be extended to the non-exclusive parametrization case.
provides noisy estimates of  $f^{*}(\pmb{x})$ , a classical computer solves the minimization problem (1) and finds the minimizer  $\hat{\pmb{x}}$ .
Several approaches, including stochastic gradient descent (SGD) [9, 10], SMO [11], and BO [15], have been proposed for solving the VQE optimization problem (1). Among others, state-of-the-art methods effectively incorporate unique properties of VQE. Let  $\{e_d\}_{d=1}^D$  be the standard basis.
Proposition 1. [9] (Parameter shift rule) The VQE objective function  $f^{*}(\cdot)$  in Eq. (6) for any parametric quantum circuit  $G(\cdot)$ , Hermitian operator  $H$ , and initial state  $|\psi_0\rangle$  satisfies
$$
2 \frac {\partial}{\partial x _ {d}} f ^ {*} (\boldsymbol {x}) = f ^ {*} \left(\boldsymbol {x} + \frac {\pi}{2} \boldsymbol {e} _ {d}\right) - f ^ {*} \left(\boldsymbol {x} - \frac {\pi}{2} \boldsymbol {e} _ {d}\right), \quad \forall \boldsymbol {x} \in [ 0, 2 \pi) ^ {D}, d = 1, \dots , D. \tag {7}
$$
Most SGD approaches rely on Proposition 1, which allows accurate gradient estimation from  $2 \cdot D$  observations. Another useful property is used for tailoring SMO [12] to VQE:
Proposition 2. [11] For the VQE objective function  $f^{*}(\cdot)$  in Eq. (6) with any  $G(\cdot), H$ , and  $|\psi_0\rangle$ ,
$$
\exists \boldsymbol {b} \in \mathbb {R} ^ {3 ^ {D}} \quad \text {s u c h t h a t} \quad f ^ {*} (\boldsymbol {x}) = \boldsymbol {b} ^ {\top} \cdot \mathbf {v e c} \left(\otimes_ {d = 1} ^ {D} (1, \cos x _ {d}, \sin x _ {d}) ^ {\top}\right), \quad \forall \boldsymbol {x} \in [ 0, 2 \pi) ^ {D}, \tag {8}
$$
where  $\otimes$  and  $\mathbf{vec}(\cdot)$  denote the tensor product and the vectorization operator for a tensor, respectively.
Proposition 2 provides a strong prior knowledge on the VQE objective function  $f^{*}(\cdot)$  — if we fix the function values at three general points along a coordinate axis, e.g.,  $\{x, x \pm \frac{2\pi}{3} e_d\}$ , for any  $x$  and  $d$ , then the whole function along the axis, i.e.,  $f^{*}(x + \alpha e_{d}), \forall \alpha \in [0, 2\pi)$  is fixed because it is a first-order sinusoidal function. Leveraging this property, a version of SMO was proposed [11]. This optimization strategy, named after its authors Nakanishi-Fuji-Todo (NFT), finds the global optimum in a single direction by sinusoidal fitting from three function values (two observed and one estimated) at each iteration, and showed state-of-the-art performance (see Appendix F.1 for details of NFT).
In this section, we introduce our approach that combines BO and NFT by using a novel kernel and a novel acquisition function. After introducing these two ingredients, we propose our approach for VQE optimization. Lastly, we prove the equivalence between Propositions 1 and 2. We refer the reader to Appendix F, in particular Algorithms 1 to 3, for the pseudo-codes and further algorithmic details complementing the brief introduction to the algorithms presented in the following sections.
3.1 VQE Kernel
We first propose the following VQE kernel, and use it for the GP regression model (2):
$$
k ^ {\mathrm {V Q E}} \left(\boldsymbol {x}, \boldsymbol {x} ^ {\prime}\right) = \sigma_ {0} ^ {2} \prod_ {d = 1} ^ {D} \left(\frac {\gamma^ {2} + 2 \cos \left(x _ {d} - x _ {d} ^ {\prime}\right)}{\gamma^ {2} + 2}\right), \tag {9}
$$
where  $\sigma_0^2, \gamma^2 > 0$  are kernel parameters.  $\sigma_0^2$  corresponds to the prior variance, while  $\gamma^2$  controls the smoothness of the kernel. For  $\gamma^2 = 1$ , the VQE kernel is the product of Dirichlet kernels [37], each of which is associated with each direction  $d$ . We can show that this kernel (9) exactly gives the finite-dimensional feature space specified by Proposition 2 (the proof is given in Appendix D):
Theorem 1. The VQE kernel (9) is decomposed as
$$
k ^ {V Q E} (\pmb {x}, \pmb {x} ^ {\prime}) = \phi (\pmb {x}) ^ {\top} \phi (\pmb {x} ^ {\prime}), w h e r e \phi (\pmb {x}) = \frac {\sigma_ {0}}{(\gamma^ {2} + 2) ^ {D / 2}} \mathbf {v e c} \left(\otimes_ {d = 1} ^ {D} (\gamma , \sqrt {2} \cos x _ {d}, \sqrt {2} \sin x _ {d}) ^ {\top}\right).
$$
Let  $\mathcal{F}^{\mathrm{VQE}}$  be the set of all possible VQE objective functions specified by Proposition 2. Theorem 1 guarantees that the support of the GP prior in Eq. (2) matches  $\mathcal{F}^{\mathrm{VQE}}$ , if we use the VQE kernel function (9) with a prior mean function such that  $\nu(\cdot) \in \mathcal{F}^{\mathrm{VQE}}$ . Thus, the VQE kernel drastically limits the expressivity of GP without model misspecification, which enhances statistical efficiency. A more important consequence of Theorem 1 is that Proposition 2 holds for any sample from the GP posterior with the VQE kernel (9). This implies that if GP is certain about three general points along an axis, it must be certain about the whole 1-D subspace going through those points. Theorem 1 can be generalized to the non-exclusive case, where each entry of  $x$  may parametrize multiple gates simultaneously (see Appendix E).
3.2 EMICoRe: Expected Maximum Improvement over Confident Regions
In GP regression, the predictive covariance does not depend on the observations  $\mathbf{y}$  (see Eq. (4)), implying that, for given training inputs  $\mathbf{X} \in \mathcal{X}^N$ , we can compute the predictive variance  $s_{\mathbf{X}}(\mathbf{x}, \mathbf{x})$  at any  $\mathbf{x} \in \mathcal{X}$  before observing the function values. Let us define Confident Regions (CoRe) as
$$
\mathcal {Z} _ {\boldsymbol {X}} = \left\{\boldsymbol {x} \in \mathcal {X}; s _ {\boldsymbol {X}} (\boldsymbol {x}, \boldsymbol {x}) \leq \kappa^ {2} \right\}, \tag {10}
$$