Enhancing the expressivity of quantum neural networks with residual connections

In the recent noisy intermediate-scale quantum era, the research on the combination of artificial intelligence and quantum computing has been greatly developed. Inspired by neural networks, developing quantum neural networks with specific structures is one of the most promising directions for improving network performance. In this work, we propose a quantum circuit-based algorithm to implement quantum residual neural networks (QResNets), where the residual connection channels are constructed by introducing auxiliary qubits to the data-encoding and trainable blocks of the quantum neural networks. Importantly, we prove that when this particular network architecture is applied to a $l$-layer data-encoding, the number of frequency generation forms can be extended from one, namely the difference of the sum of generator eigenvalues, to $\mathcal{O}(l^2)$. And the flexibility in adjusting the corresponding Fourier coefficients can also be improved due to the diversity of spectrum construction methods and the additional optimization degrees of freedom in the generalized residual operators. These results indicate that the residual encoding scheme can achieve better spectral richness and enhance the expressivity of various parameterized quantum circuits. Extensive numerical demonstrations in regression tasks of fitting various functions and applications in image classification with MNIST datasets are offered to present the expressivity enhancement. Our work lays the foundation for a complete quantum implementation of the classical residual neural networks and explores a new strategy for quantum feature map in quantum machine learning.