  Quantum neural networks (QNNs) have become an important tool for
understanding the physical world, but their advantages and limitations are not
fully understood. Some QNNs with specific encoding methods can be efficiently
simulated by classical surrogates, while others with quantum memory may perform
better than classical classifiers. Here we systematically investigate the
problem-dependent power of quantum neural classifiers (QCs) on multi-class
classification tasks. Through the analysis of expected risk, a measure that
weighs the training loss and the generalization error of a classifier jointly,
we identify two key findings: first, the training loss dominates the power
rather than the generalization ability; second, QCs undergo a U-shaped risk
curve, in contrast to the double-descent risk curve of deep neural classifiers.
We also reveal the intrinsic connection between optimal QCs and the Helstrom
bound and the equiangular tight frame. Using these findings, we propose a
method that uses loss dynamics to probe whether a QC may be more effective than
a classical classifier on a particular learning task. Numerical results
demonstrate the effectiveness of our approach to explain the superiority of QCs
over multilayer Perceptron on parity datasets and their limitations over
convolutional neural networks on image datasets. Our work sheds light on the
problem-dependent power of QNNs and offers a practical tool for evaluating
their potential merit.
