This paper proposes a novel Bayesian optimization method for Variational Quantum Eigensolvers (VQEs), combining physics-informed priors with Bayesian optimization. The method introduces a VQE-kernel that aligns with the functional form of the VQE objective function, reducing posterior uncertainty and improving statistical efficiency. A new acquisition function, EMICoRe, is introduced to exploit the inductive bias of the VQE-kernel by treating low-uncertainty regions as "observed." This allows the determination of the complete objective function along a one-dimensional subspace with as few as three observations. The method also integrates the Nakanishi-Fuji-Todo (NFT) approach, which uses sinusoidal fitting to find the global optimum in a single direction. The combination of NFT and EMICoRe enhances the efficiency of NFT by replacing sub-optimal deterministic choices with Bayesian optimization, while NFT constrains the exploration space of BO, leading to significant improvements in scalability. Numerical experiments demonstrate that the proposed method outperforms state-of-the-art baselines in VQE problems with different Hamiltonians and qubit numbers. The paper also proves that two known properties of VQEs—the parameter shift rule and the sinusoidal function-form—are mathematically equivalent, indicating they are two expressions of a single property. The method is shown to be effective in optimizing VQEs with different levels of observation noise, making it suitable for efficient VQE optimization in the NISQ era.This paper proposes a novel Bayesian optimization method for Variational Quantum Eigensolvers (VQEs), combining physics-informed priors with Bayesian optimization. The method introduces a VQE-kernel that aligns with the functional form of the VQE objective function, reducing posterior uncertainty and improving statistical efficiency. A new acquisition function, EMICoRe, is introduced to exploit the inductive bias of the VQE-kernel by treating low-uncertainty regions as "observed." This allows the determination of the complete objective function along a one-dimensional subspace with as few as three observations. The method also integrates the Nakanishi-Fuji-Todo (NFT) approach, which uses sinusoidal fitting to find the global optimum in a single direction. The combination of NFT and EMICoRe enhances the efficiency of NFT by replacing sub-optimal deterministic choices with Bayesian optimization, while NFT constrains the exploration space of BO, leading to significant improvements in scalability. Numerical experiments demonstrate that the proposed method outperforms state-of-the-art baselines in VQE problems with different Hamiltonians and qubit numbers. The paper also proves that two known properties of VQEs—the parameter shift rule and the sinusoidal function-form—are mathematically equivalent, indicating they are two expressions of a single property. The method is shown to be effective in optimizing VQEs with different levels of observation noise, making it suitable for efficient VQE optimization in the NISQ era.