  As a hybrid of artificial intelligence and quantum computing, quantum neural
networks (QNNs) have gained significant attention as a promising application on
near-term, noisy intermediate-scale quantum (NISQ) devices. Conventional QNNs
are described by parametrized quantum circuits, which perform unitary
operations and measurements on quantum states. In this work, we propose a novel
approach to enhance the expressivity of QNNs by incorporating randomness into
quantum circuits. Specifically, we introduce a random layer, which contains
single-qubit gates sampled from an trainable ensemble pooling. The prediction
of QNN is then represented by an ensemble average over a classical function of
measurement outcomes. We prove that our approach can accurately approximate
arbitrary target operators using Uhlmann's theorem for majorization, which
enables observable learning. Our proposal is demonstrated with extensive
numerical experiments, including observable learning, R\'enyi entropy
measurement, and image recognition. We find the expressivity of QNNs is
enhanced by introducing randomness for multiple learning tasks, which could
have broad application in quantum machine learning.
