General Vapnik–Chervonenkis dimension bounds for quantum circuit learning

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.