  Classification of quantum data is essential for quantum machine learning and
near-term quantum technologies. In this paper, we propose a new hybrid
quantum-classical framework for supervised quantum learning, which we call
Variational Shadow Quantum Learning (VSQL). Our method in particular utilizes
the classical shadows of quantum data, which fundamentally represent the side
information of quantum data with respect to certain physical observables.
Specifically, we first use variational shadow quantum circuits to extract
classical features in a convolution way and then utilize a fully-connected
neural network to complete the classification task. We show that this method
could sharply reduce the number of parameters and thus better facilitate
quantum circuit training. Simultaneously, less noise will be introduced since
fewer quantum gates are employed in such shadow circuits. Moreover, we show
that the Barren Plateau issue, a significant gradient vanishing problem in
quantum machine learning, could be avoided in VSQL. Finally, we demonstrate the
efficiency of VSQL in quantum classification via numerical experiments on the
classification of quantum states and the recognition of multi-labeled
handwritten digits. In particular, our VSQL approach outperforms existing
variational quantum classifiers in the test accuracy in the binary case of
handwritten digit recognition and notably requires much fewer parameters.
