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Quantum computing has emerged as a powerful computational paradigm capable of solving problems beyond the reach of classical computers. However, today's quantum computers are noisy, posing challenges to obtaining accurate results. Here, we explore the impact of noise on quantum computing, focusing on the challenges in sampling bit strings from noisy quantum computers and the implications for optimization and machine learning. We formally quantify the sampling overhead to extract good samples from noisy quantum computers and relate it to the layer fidelity, a metric to determine the performance of noisy quantum processors. Further, we show how this allows us to use the conditional value at risk of noisy samples to determine provable bounds on noise-free expectation values. We discuss how to leverage these bounds for different algorithms and demonstrate our findings through experiments on real quantum computers involving up to 127 qubits. The results show strong alignment with theoretical predictions., Competing Interests: Competing interests: The authors declare no competing interests., (© 2024. IBM.)

It is unknown whether near-term quantum computers are advantageous for machine learning tasks. In this work we address this question by trying to understand how powerful and trainable quantum machine learning models are in relation to popular classical neural networks. We propose the effective dimension-a measure that captures these qualities-and prove that it can be used to assess any statistical model's ability to generalize on new data. Crucially, the effective dimension is a data-dependent measure that depends on the Fisher information, which allows us to gauge the ability of a model to train. We demonstrate numerically that a class of quantum neural networks is able to achieve a considerably better effective dimension than comparable feedforward networks and train faster, suggesting an advantage for quantum machine learning, which we verify on real quantum hardware., (© 2021. The Author(s), under exclusive licence to Springer Nature America, Inc.)