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In this work, we propose a new Gaussian process (GP) regression framework that enforces the physical constraints in a probabilistic manner. Specifically, we focus on inequality and monotonicity constraints. This GP model is trained by the quantum-inspired Hamiltonian Monte Carlo (QHMC) algorithm, which is an efficient way to sample from a broad class of distributions by allowing a particle to have a random mass matrix with a probability distribution. Integrating the QHMC into the inequality and monotonicity constrained GP regression in the probabilistic sense, our approach enhances the accuracy and reduces the variance in the resulting GP model. Additionally, the probabilistic aspect of the method leads to reduced computational expenses and execution time. Further, we present an adaptive learning algorithm that guides the selection of constraint locations. The accuracy and efficiency of the method are demonstrated in estimating the hyperparameter of high-dimensional GP models under noisy conditions, reconstructing the sparsely observed state of a steady state heat transport problem, and learning a conservative tracer distribution from sparse tracer concentration measurements.