Combinatorial optimization problems are ubiquitous and computationally hard to solve in general. Quantum computing is envisioned as a powerful tool offering potential computational advantages for solving some of these problems. Quantum approximate optimization algorithm (QAOA), one of the most representative quantum-classical hybrid algorithms, is designed to solve certain combinatorial optimization problems by transforming a discrete optimization problem into a classical optimization problem over a continuous circuit parameter domain. QAOA objective landscape over the parameter variables is notorious for pervasive local minima and barren plateaus, and its viability in training significantly relies on the efficacy of the classical optimization algorithm. To enhance the performance of QAOA, we design double adaptive-region Bayesian optimization (DARBO), an adaptive classical optimizer for QAOA. Our experimental results demonstrate that the algorithm greatly outperforms conventional gradient-based and gradient-free optimizers in terms of speed, accuracy, and stability. We also address the issues of measurement efficiency and the suppression of quantum noise by successfully conducting the full optimization loop on the superconducting quantum processor. This work helps to unlock the full power of QAOA and paves the way toward achieving quantum advantage in practical classical tasks.

We introduce an unsupervised clustering algorithm to improve training efficiency and accuracy in predicting energies using molecular-orbital-based machine learning (MOB-ML). This work determines clusters via the Gaussian mixture model (GMM) in an entirely automatic manner and simplifies an earlier supervised clustering approach [J. Chem. Theory Comput., 15, 6668 (2019)] by eliminating both the necessity for user-specified parameters and the training of an additional classifier. Unsupervised clustering results from GMM have the advantage of accurately reproducing chemically intuitive groupings of frontier molecular orbitals and having improved performance with an increasing number of training examples. The resulting clusters from supervised or unsupervised clustering is further combined with scalable Gaussian process regression (GPR) or linear regression (LR) to learn molecular energies accurately by generating a local regression model in each cluster. Among all four combinations of regressors and clustering methods, GMM combined with scalable exact Gaussian process regression (GMM/GPR) is the most efficient training protocol for MOB-ML. The numerical tests of molecular energy learning on thermalized datasets of drug-like molecules demonstrate the improved accuracy, transferability, and learning efficiency of GMM/GPR over not only other training protocols for MOB-ML, i.e., supervised regression-clustering combined with GPR(RC/GPR) and GPR without clustering. GMM/GPR also provide the best molecular energy predictions compared with the ones from literature on the same benchmark datasets. With a lower scaling, GMM/GPR has a 10.4-fold speedup in wall-clock training time compared with scalable exact GPR with a training size of 6500 QM7b-T molecules.

We present an application of the blackbox matrix-matrix multiplication (BBMM) algorithm to scale up the Gaussian Process (GP) training of molecular energies in the molecular-orbital based machine learning (MOB-ML) framework. An alternative implementation of BBMM (AltBBMM) is also proposed to train more efficiently (over four-fold speedup) with the same accuracy and transferability as the original BBMM implementation. The training of MOB-ML was limited to 220 molecules, and BBMM and AltBBMM scale the training of MOB-ML up by over 30 times to 6500 molecules (more than a million pair energies). The accuracy and transferability of both algorithms are examined on the benchmark datasets of organic molecules with 7 and 13 heavy atoms. These lower-scaling implementations of the GP preserve the state-of-the-art learning efficiency in the low-data regime while extending it to the large-data regime with better accuracy than other available machine learning works on molecular energies.

Machine learning (ML) in the representation of molecular-orbital-based (MOB) features has been shown to be an accurate and transferable approach to the prediction of post-Hartree-Fock correlation energies. Previous applications of MOB-ML employed Gaussian Process Regression (GPR), which provides good prediction accuracy with small training sets; however, the cost of GPR training scales cubically with the amount of data and becomes a computational bottleneck for large training sets. In the current work, we address this problem by introducing a clustering/regression/classification implementation of MOB-ML. In a first step, regression clustering (RC) is used to partition the training data to best fit an ensemble of linear regression (LR) models; in a second step, each cluster is regressed independently, using either LR or GPR; and in a third step, a random forest classifier (RFC) is trained for the prediction of cluster assignments based on MOB feature values. Upon inspection, RC is found to recapitulate chemically intuitive groupings of the frontier molecular orbitals, and the combined RC/LR/RFC and RC/GPR/RFC implementations of MOB-ML are found to provide good prediction accuracy with greatly reduced wall-clock training times. For a dataset of thermalized geometries of 7211 organic molecules of up to seven heavy atoms, both implementations reach chemical accuracy (1 kcal/mol error) with only 300 training molecules, while providing 35000-fold and 4500-fold reductions in the wall-clock training time, respectively, compared to MOB-ML without clustering. The resulting models are also demonstrated to retain transferability for the prediction of large-molecule energies with only small-molecule training data. Finally, it is shown that capping the number of training datapoints per cluster leads to further improvements in prediction accuracy with negligible increases in wall-clock training time.