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Quantum computing has the potential to outperform classical computers and is expected to play an active role in various fields. In quantum machine learning, a quantum computer has been found useful for enhanced feature representation and high-dimensional state or function approximation. Quantum–classical hybrid algorithms have been proposed in recent years for this purpose under the noisy intermediate-scale quantum computer (NISQ) environment. Under this scheme, the role played by the classical computer is the parameter tuning, parameter optimization, and parameter update for the quantum circuit. In this paper, we propose a gradient descent-based backpropagation algorithm that can efficiently calculate the gradient in parameter optimization and update the parameter for quantum circuit learning, which outperforms the current parameter search algorithms in terms of computing speed while presenting the same or even higher test accuracy. Meanwhile, the proposed theoretical scheme was successfully implemented on the 20-qubit quantum computer of IBM Q, ibmq_johannesburg. The experimental results reveal that the gate error, especially the CNOT gate error, strongly affects the derived gradient accuracy. The regression accuracy performed on the IBM Q becomes lower with the increase in the number of measurement shot times due to the accumulated gate noise error.

Quantifying the model complexity of quantum circuits provides a guide to avoid overfitting in quantum machine learning. Previously we established a Vapnik–Chervonenkis (VC) dimension upper bound for ‘encoding-first’ quantum circuits, where the input layer is the first layer of the circuit. In this work, we prove a general VC dimension upper bound for quantum circuit learning including ‘data re-uploading’ circuits, where the input gates can be single qubit rotations anywhere in the circuit. A linear lower bound is also constructed. The properties of the bounds and approximation-estimation trade-off considerations are discussed.