The Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) is one of the most advanced algorithms in numerical black-box optimization. For noisy objective functions, several approaches were proposed to mitigate the noise, e.g., re-evaluations of the same solution or adapting the population size. In this paper, we devise a novel method to adaptively choose the optimal re-evaluation number for function values corrupted by additive Gaussian white noise. We derive a theoretical lower bound of the expected improvement achieved in one iteration of CMA-ES, given an estimation of the noise level and the Lipschitz constant of the function's gradient. Solving for the maximum of the lower bound, we obtain a simple expression of the optimal re-evaluation number. We experimentally compare our method to the state-of-the-art noise-handling methods for CMA-ES on a set of artificial test functions across various noise levels, optimization budgets, and dimensionality. Our method demonstrates significant advantages in terms of the probability of hitting near-optimal function values.

Variational quantum algorithms, such as the Recursive Quantum Approximate Optimization Algorithm (RQAOA), have become increasingly popular, offering promising avenues for employing Noisy Intermediate-Scale Quantum devices to address challenging combinatorial optimization tasks like the maximum cut problem. In this study, we utilize an evolutionary algorithm equipped with a unique fitness function. This approach targets hard maximum cut instances within the latent space of a Graph Autoencoder, identifying those that pose significant challenges or are particularly tractable for RQAOA, in contrast to the classic Goemans and Williamson algorithm. Our findings not only delineate the distinct capabilities and limitations of each algorithm but also expand our understanding of RQAOA's operational limits. Furthermore, the diverse set of graphs we have generated serves as a crucial benchmarking asset, emphasizing the need for more advanced algorithms to tackle combinatorial optimization challenges. Additionally, our results pave the way for new avenues in graph generation research, offering exciting opportunities for future explorations.

The key challenge in the noisy intermediate-scale quantum era is finding useful circuits compatible with current device limitations. Variational quantum algorithms (VQAs) offer a potential solution by fixing the circuit architecture and optimizing individual gate parameters in an external loop. However, parameter optimization can become intractable, and the overall performance of the algorithm depends heavily on the initially chosen circuit architecture. Several quantum architecture search (QAS) algorithms have been developed to design useful circuit architectures automatically. In the case of parameter optimization alone, noise effects have been observed to dramatically influence the performance of the optimizer and final outcomes, which is a key line of study. However, the effects of noise on the architecture search, which could be just as critical, are poorly understood. This work addresses this gap by introducing a curriculum-based reinforcement learning QAS (CRLQAS) algorithm designed to tackle challenges in realistic VQA deployment. The algorithm incorporates (i) a 3D architecture encoding and restrictions on environment dynamics to explore the search space of possible circuits efficiently, (ii) an episode halting scheme to steer the agent to find shorter circuits, and (iii) a novel variant of simultaneous perturbation stochastic approximation as an optimizer for faster convergence. To facilitate studies, we developed an optimized simulator for our algorithm, significantly improving computational efficiency in simulating noisy quantum circuits by employing the Pauli-transfer matrix formalism in the Pauli-Liouville basis. Numerical experiments focusing on quantum chemistry tasks demonstrate that CRLQAS outperforms existing QAS algorithms across several metrics in both noiseless and noisy environments.

Abstract As restricted quantum computers become available, research focuses on finding meaningful applications. For example, in quantum machine learning, a special type of quantum circuit called a quantum neural network is one of the most investigated approaches. However, we know little about suitable circuit architectures or important model hyperparameters for a given task. In this work, we apply the functional ANOVA framework to the quantum neural network architectures to analyze which of the quantum machine learning hyperparameters are most influential for their predictive performance. We restrict our study to 7 open-source datasets from the OpenML-CC18 classification benchmark, which are small enough for simulations on quantum hardware with fewer than 20 qubits. Using this framework, three main levels of importance were identified, confirming expected patterns and revealing new insights. For instance, the learning rate is identified as the most important hyperparameter on all datasets, whereas the particular choice of entangling gates used is found to be the least important on all except for one dataset. In addition to identifying the relevant hyperparameters, for each of them, we also learned data-driven priors based on values that perform well on previously seen datasets, which can then be used to steer hyperparameter optimization processes. We utilize these priors in the hyperparameter optimization method hyperband and show that these improve performance against uniform sampling across all datasets by, on average, $$0.53 \%$$ 0.53 % , up to $$6.11 \%$$ 6.11 % , in cross-validation accuracy. We also demonstrate that such improvements hold on average regardless of the configuration hyperband is run with. Our work introduces new methodologies for studying quantum machine learning models toward quantum model selection in practice. All research code is made publicly available.